Increasing and decreasing intervals show where a function is going up or going down as \(x\) increases. To find these intervals, take the first derivative and determine where \(f'(x) \gt 0\) or \(f'(x) \lt 0\). Critical numbers happen where \(f'(x)=0\), where \(f'(x)\) is undefined, or at endpoints of the domain.
Notes
Critical Numbers
\(\bullet\, f'(x)= 0\)
\(\bullet\, f'(x)\) is undefined
\(\bullet\, \) Endpoints of domain
Increasing/Decreasing Intervals
Increasing when \(f'(x)\gt 0\)
Decreasing when \(f'(x)\lt 0\)
Practice Problems
For each function
a. Find the critical numbers
b. Find the open intervals where f is increasing
c. Find the open intervals where f is decreasing
\(\textbf{1)}\) \( f(x)=2x^2+4x+3 \)
The critical number is \( -1 \)
The increasing interval is \((-1,\infty) \)
The decreasing interval is \( (-\infty,-1) \)
\(\,\,\,\,\,\,f(x)=2x^2+4x+3\)
\(\,\,\,\,\,\,f'(x)=4x+4\)
\(\,\,\,\,\,\,4x+4=0\)
\(\,\,\,\,\,\,x=-1\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-\infty,-1)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-1,\infty)\)
\(\,\,\,\,\,\)The critical number is \(-1\). The function is increasing on \((-1,\infty)\) and decreasing on \((-\infty,-1)\).
\(\,\,\,\,\,\,f(x)=2x^2+4x+3\)
\(\,\,\,\,\,\,f'(x)=4x+4\)
\(\,\,\,\,\,\,4x+4=0\)
\(\,\,\,\,\,\,x=-1\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-\infty,-1)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-1,\infty)\)
\(\,\,\,\,\,\)The critical number is \(-1\). The function is increasing on \((-1,\infty)\) and decreasing on \((-\infty,-1)\).
\(\textbf{2)}\) \( f(x)=\frac{1}{5}x^5-16x+5 \)
The critical numbers are \( \pm 2 \)
The increasing interval is \( (- \infty,-2) \cup (2, \infty) \)
The decreasing interval is \( (-2,2) \)
\(\,\,\,\,\,\,f(x)=\frac{1}{5}x^5-16x+5\)
\(\,\,\,\,\,\,f'(x)=x^4-16\)
\(\,\,\,\,\,\,x^4-16=0\)
\(\,\,\,\,\,\,(x^2-4)(x^2+4)=0\)
\(\,\,\,\,\,\,x=-2\text{ or }x=2\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,-2)\cup(2,\infty)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-2,2)\)
\(\,\,\,\,\,\)The critical numbers are \(\pm2\). The function is increasing on \((-\infty,-2)\cup(2,\infty)\) and decreasing on \((-2,2)\).
\(\,\,\,\,\,\,f(x)=\frac{1}{5}x^5-16x+5\)
\(\,\,\,\,\,\,f'(x)=x^4-16\)
\(\,\,\,\,\,\,x^4-16=0\)
\(\,\,\,\,\,\,(x^2-4)(x^2+4)=0\)
\(\,\,\,\,\,\,x=-2\text{ or }x=2\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,-2)\cup(2,\infty)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-2,2)\)
\(\,\,\,\,\,\)The critical numbers are \(\pm2\). The function is increasing on \((-\infty,-2)\cup(2,\infty)\) and decreasing on \((-2,2)\).
\(\textbf{3)}\) \( f(x)=-3x+2 \)
There are no critical numbers.
The increasing interval is \( \emptyset \)
The decreasing interval is \( (- \infty, \infty) \)
\(\,\,\,\,\,\,f(x)=-3x+2\)
\(\,\,\,\,\,\,f'(x)=-3\)
\(\,\,\,\,\,\,-3 \lt 0\text{ for all }x\)
\(\,\,\,\,\,\,\text{The derivative is never }0\text{ or undefined.}\)
\(\,\,\,\,\,\)There are no critical numbers. The function is decreasing on \((-\infty,\infty)\) and increasing on \(\emptyset\).
\(\,\,\,\,\,\,f(x)=-3x+2\)
\(\,\,\,\,\,\,f'(x)=-3\)
\(\,\,\,\,\,\,-3 \lt 0\text{ for all }x\)
\(\,\,\,\,\,\,\text{The derivative is never }0\text{ or undefined.}\)
\(\,\,\,\,\,\)There are no critical numbers. The function is decreasing on \((-\infty,\infty)\) and increasing on \(\emptyset\).
\(\textbf{4)}\) \( f(x)=4x^3+1 \)
The critical number is \( 0 \) (plateau)
The increasing interval is \( (-\infty,\infty) \)
The decreasing interval is \( \emptyset \)
\(\,\,\,\,\,\,f(x)=4x^3+1\)
\(\,\,\,\,\,\,f'(x)=12x^2\)
\(\,\,\,\,\,\,12x^2=0\)
\(\,\,\,\,\,\,x=0\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,0)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(0,\infty)\)
\(\,\,\,\,\,\)The critical number is \(0\). The function is increasing on \((-\infty,\infty)\) and decreasing on \(\emptyset\).
\(\,\,\,\,\,\,f(x)=4x^3+1\)
\(\,\,\,\,\,\,f'(x)=12x^2\)
\(\,\,\,\,\,\,12x^2=0\)
\(\,\,\,\,\,\,x=0\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,0)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(0,\infty)\)
\(\,\,\,\,\,\)The critical number is \(0\). The function is increasing on \((-\infty,\infty)\) and decreasing on \(\emptyset\).
\(\textbf{5)}\) \( f(x)=x^3-3x^2 \)
The critical numbers are \(0\) and \(2\)
The increasing interval is \((-\infty,0)\cup(2,\infty)\)
The decreasing interval is \((0,2)\)
\(\,\,\,\,\,\,f(x)=x^3-3x^2\)
\(\,\,\,\,\,\,f'(x)=3x^2-6x\)
\(\,\,\,\,\,\,3x^2-6x=0\)
\(\,\,\,\,\,\,3x(x-2)=0\)
\(\,\,\,\,\,\,x=0\text{ or }x=2\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,0)\cup(2,\infty)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(0,2)\)
\(\,\,\,\,\,\)The critical numbers are \(0\) and \(2\).
\(\,\,\,\,\,\,f(x)=x^3-3x^2\)
\(\,\,\,\,\,\,f'(x)=3x^2-6x\)
\(\,\,\,\,\,\,3x^2-6x=0\)
\(\,\,\,\,\,\,3x(x-2)=0\)
\(\,\,\,\,\,\,x=0\text{ or }x=2\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,0)\cup(2,\infty)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(0,2)\)
\(\,\,\,\,\,\)The critical numbers are \(0\) and \(2\).
\(\textbf{6)}\) \( f(x)=x^4-8x^2 \)
The critical numbers are \(-2,0,2\)
The increasing interval is \((-2,0)\cup(2,\infty)\)
The decreasing interval is \((-\infty,-2)\cup(0,2)\)
\(\,\,\,\,\,\,f(x)=x^4-8x^2\)
\(\,\,\,\,\,\,f'(x)=4x^3-16x\)
\(\,\,\,\,\,\,4x^3-16x=0\)
\(\,\,\,\,\,\,4x(x^2-4)=0\)
\(\,\,\,\,\,\,4x(x-2)(x+2)=0\)
\(\,\,\,\,\,\,x=-2,0,2\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-\infty,-2)\cup(0,2)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-2,0)\cup(2,\infty)\)
\(\,\,\,\,\,\)The critical numbers are \(-2,0,2\).
\(\,\,\,\,\,\,f(x)=x^4-8x^2\)
\(\,\,\,\,\,\,f'(x)=4x^3-16x\)
\(\,\,\,\,\,\,4x^3-16x=0\)
\(\,\,\,\,\,\,4x(x^2-4)=0\)
\(\,\,\,\,\,\,4x(x-2)(x+2)=0\)
\(\,\,\,\,\,\,x=-2,0,2\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-\infty,-2)\cup(0,2)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-2,0)\cup(2,\infty)\)
\(\,\,\,\,\,\)The critical numbers are \(-2,0,2\).
\(\textbf{7)}\) \( f(x)=x^4-4x^3 \)
The critical numbers are \(0\) and \(3\)
The increasing interval is \((3,\infty)\)
The decreasing interval is \((-\infty,0)\cup(0,3)\)
\(\,\,\,\,\,\,f(x)=x^4-4x^3\)
\(\,\,\,\,\,\,f'(x)=4x^3-12x^2\)
\(\,\,\,\,\,\,4x^3-12x^2=0\)
\(\,\,\,\,\,\,4x^2(x-3)=0\)
\(\,\,\,\,\,\,x=0\text{ or }x=3\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-\infty,0)\cup(0,3)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(3,\infty)\)
\(\,\,\,\,\,\)The critical numbers are \(0\) and \(3\).
\(\,\,\,\,\,\,f(x)=x^4-4x^3\)
\(\,\,\,\,\,\,f'(x)=4x^3-12x^2\)
\(\,\,\,\,\,\,4x^3-12x^2=0\)
\(\,\,\,\,\,\,4x^2(x-3)=0\)
\(\,\,\,\,\,\,x=0\text{ or }x=3\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-\infty,0)\cup(0,3)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(3,\infty)\)
\(\,\,\,\,\,\)The critical numbers are \(0\) and \(3\).
\(\textbf{8)}\) \( f(x)=\frac{x+1}{x-2} \)
There are no critical numbers. \(x=2\) is not in the domain.
The increasing interval is \(\emptyset\)
The decreasing intervals are \((-\infty,2)\) and \((2,\infty)\)
\(\,\,\,\,\,\,f(x)=\frac{x+1}{x-2}\)
\(\,\,\,\,\,\,f'(x)=\frac{(x-2)(1)-(x+1)(1)}{(x-2)^2}\)
\(\,\,\,\,\,\,f'(x)=\frac{x-2-x-1}{(x-2)^2}\)
\(\,\,\,\,\,\,f'(x)=\frac{-3}{(x-2)^2}\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-\infty,2)\text{ and }(2,\infty)\)
\(\,\,\,\,\,\,x=2\text{ is not in the domain, so it is not a critical number.}\)
\(\,\,\,\,\,\)The function is decreasing on \((-\infty,2)\) and \((2,\infty)\).
\(\,\,\,\,\,\,f(x)=\frac{x+1}{x-2}\)
\(\,\,\,\,\,\,f'(x)=\frac{(x-2)(1)-(x+1)(1)}{(x-2)^2}\)
\(\,\,\,\,\,\,f'(x)=\frac{x-2-x-1}{(x-2)^2}\)
\(\,\,\,\,\,\,f'(x)=\frac{-3}{(x-2)^2}\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-\infty,2)\text{ and }(2,\infty)\)
\(\,\,\,\,\,\,x=2\text{ is not in the domain, so it is not a critical number.}\)
\(\,\,\,\,\,\)The function is decreasing on \((-\infty,2)\) and \((2,\infty)\).
\(\textbf{9)}\) \( f(x)=\sqrt{x+4} \)
The critical number is \(-4\)
The increasing interval is \((-4,\infty)\)
The decreasing interval is \(\emptyset\)
\(\,\,\,\,\,\,f(x)=\sqrt{x+4}\)
\(\,\,\,\,\,\,f'(x)=\frac{1}{2\sqrt{x+4}}\)
\(\,\,\,\,\,\,f'(x)\text{ is undefined at }x=-4\)
\(\,\,\,\,\,\,x=-4\text{ is in the domain of }f(x)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-4,\infty)\)
\(\,\,\,\,\,\)The critical number is \(-4\). The function is increasing on \((-4,\infty)\).
\(\,\,\,\,\,\,f(x)=\sqrt{x+4}\)
\(\,\,\,\,\,\,f'(x)=\frac{1}{2\sqrt{x+4}}\)
\(\,\,\,\,\,\,f'(x)\text{ is undefined at }x=-4\)
\(\,\,\,\,\,\,x=-4\text{ is in the domain of }f(x)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-4,\infty)\)
\(\,\,\,\,\,\)The critical number is \(-4\). The function is increasing on \((-4,\infty)\).
\(\textbf{10)}\) \( f(x)=\sqrt[3]{x} \)
The critical number is \(0\)
The increasing interval is \((-\infty,0)\cup(0,\infty)\)
The decreasing interval is \(\emptyset\)
\(\,\,\,\,\,\,f(x)=\sqrt[3]{x}=x^{1/3}\)
\(\,\,\,\,\,\,f'(x)=\frac{1}{3}x^{-2/3}\)
\(\,\,\,\,\,\,f'(x)=\frac{1}{3x^{2/3}}\)
\(\,\,\,\,\,\,f'(x)\text{ is undefined at }x=0\)
\(\,\,\,\,\,\,x=0\text{ is in the domain of }f(x)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,0)\cup(0,\infty)\)
\(\,\,\,\,\,\)The critical number is \(0\). The function is increasing on \((-\infty,0)\cup(0,\infty)\).
\(\,\,\,\,\,\,f(x)=\sqrt[3]{x}=x^{1/3}\)
\(\,\,\,\,\,\,f'(x)=\frac{1}{3}x^{-2/3}\)
\(\,\,\,\,\,\,f'(x)=\frac{1}{3x^{2/3}}\)
\(\,\,\,\,\,\,f'(x)\text{ is undefined at }x=0\)
\(\,\,\,\,\,\,x=0\text{ is in the domain of }f(x)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,0)\cup(0,\infty)\)
\(\,\,\,\,\,\)The critical number is \(0\). The function is increasing on \((-\infty,0)\cup(0,\infty)\).
\(\textbf{11)}\) \( f(x)=x+\frac{9}{x} \)
The critical numbers are \(-3\) and \(3\)
The increasing intervals are \((-\infty,-3)\) and \((3,\infty)\)
The decreasing intervals are \((-3,0)\) and \((0,3)\)
\(\,\,\,\,\,\,f(x)=x+\frac{9}{x}\)
\(\,\,\,\,\,\,f'(x)=1-\frac{9}{x^2}\)
\(\,\,\,\,\,\,1-\frac{9}{x^2}=0\)
\(\,\,\,\,\,\,1=\frac{9}{x^2}\)
\(\,\,\,\,\,\,x^2=9\)
\(\,\,\,\,\,\,x=-3\text{ or }x=3\)
\(\,\,\,\,\,\,x=0\text{ is not in the domain.}\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,-3)\cup(3,\infty)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-3,0)\cup(0,3)\)
\(\,\,\,\,\,\)The critical numbers are \(-3\) and \(3\).
\(\,\,\,\,\,\,f(x)=x+\frac{9}{x}\)
\(\,\,\,\,\,\,f'(x)=1-\frac{9}{x^2}\)
\(\,\,\,\,\,\,1-\frac{9}{x^2}=0\)
\(\,\,\,\,\,\,1=\frac{9}{x^2}\)
\(\,\,\,\,\,\,x^2=9\)
\(\,\,\,\,\,\,x=-3\text{ or }x=3\)
\(\,\,\,\,\,\,x=0\text{ is not in the domain.}\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,-3)\cup(3,\infty)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-3,0)\cup(0,3)\)
\(\,\,\,\,\,\)The critical numbers are \(-3\) and \(3\).
\(\textbf{12)}\) \( f(x)=x^2e^{-x} \)
The critical numbers are \(0\) and \(2\)
The increasing interval is \((0,2)\)
The decreasing intervals are \((-\infty,0)\) and \((2,\infty)\)
\(\,\,\,\,\,\,f(x)=x^2e^{-x}\)
\(\,\,\,\,\,\,f'(x)=2xe^{-x}+x^2(-e^{-x})\)
\(\,\,\,\,\,\,f'(x)=e^{-x}(2x-x^2)\)
\(\,\,\,\,\,\,f'(x)=e^{-x}x(2-x)\)
\(\,\,\,\,\,\,e^{-x}x(2-x)=0\)
\(\,\,\,\,\,\,x=0\text{ or }x=2\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-\infty,0)\cup(2,\infty)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(0,2)\)
\(\,\,\,\,\,\)The critical numbers are \(0\) and \(2\).
\(\,\,\,\,\,\,f(x)=x^2e^{-x}\)
\(\,\,\,\,\,\,f'(x)=2xe^{-x}+x^2(-e^{-x})\)
\(\,\,\,\,\,\,f'(x)=e^{-x}(2x-x^2)\)
\(\,\,\,\,\,\,f'(x)=e^{-x}x(2-x)\)
\(\,\,\,\,\,\,e^{-x}x(2-x)=0\)
\(\,\,\,\,\,\,x=0\text{ or }x=2\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-\infty,0)\cup(2,\infty)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(0,2)\)
\(\,\,\,\,\,\)The critical numbers are \(0\) and \(2\).
\(\textbf{13)}\) \( f(x)=\ln{x}-x \)
The critical number is \(1\)
The increasing interval is \((0,1)\)
The decreasing interval is \((1,\infty)\)
\(\,\,\,\,\,\,f(x)=\ln{x}-x\)
\(\,\,\,\,\,\,f'(x)=\frac{1}{x}-1\)
\(\,\,\,\,\,\,\frac{1}{x}-1=0\)
\(\,\,\,\,\,\,\frac{1}{x}=1\)
\(\,\,\,\,\,\,x=1\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(0,1)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(1,\infty)\)
\(\,\,\,\,\,\)The critical number is \(1\).
\(\,\,\,\,\,\,f(x)=\ln{x}-x\)
\(\,\,\,\,\,\,f'(x)=\frac{1}{x}-1\)
\(\,\,\,\,\,\,\frac{1}{x}-1=0\)
\(\,\,\,\,\,\,\frac{1}{x}=1\)
\(\,\,\,\,\,\,x=1\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(0,1)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(1,\infty)\)
\(\,\,\,\,\,\)The critical number is \(1\).
\(\textbf{14)}\) \( f(x)=\sin{x}\text{ on }[0,2\pi] \)
The critical numbers are \(0,\frac{\pi}{2},\frac{3\pi}{2},2\pi\)
The increasing intervals are \(\left(0,\frac{\pi}{2}\right)\) and \(\left(\frac{3\pi}{2},2\pi\right)\)
The decreasing interval is \(\left(\frac{\pi}{2},\frac{3\pi}{2}\right)\)
\(\,\,\,\,\,\,f(x)=\sin{x}\)
\(\,\,\,\,\,\,f'(x)=\cos{x}\)
\(\,\,\,\,\,\,\cos{x}=0\)
\(\,\,\,\,\,\,x=\frac{\pi}{2},\frac{3\pi}{2}\)
\(\,\,\,\,\,\,\text{Include the endpoints of the domain: }0\text{ and }2\pi\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }\left(0,\frac{\pi}{2}\right)\cup\left(\frac{3\pi}{2},2\pi\right)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }\left(\frac{\pi}{2},\frac{3\pi}{2}\right)\)
\(\,\,\,\,\,\)The critical numbers are \(0,\frac{\pi}{2},\frac{3\pi}{2},2\pi\).
\(\,\,\,\,\,\,f(x)=\sin{x}\)
\(\,\,\,\,\,\,f'(x)=\cos{x}\)
\(\,\,\,\,\,\,\cos{x}=0\)
\(\,\,\,\,\,\,x=\frac{\pi}{2},\frac{3\pi}{2}\)
\(\,\,\,\,\,\,\text{Include the endpoints of the domain: }0\text{ and }2\pi\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }\left(0,\frac{\pi}{2}\right)\cup\left(\frac{3\pi}{2},2\pi\right)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }\left(\frac{\pi}{2},\frac{3\pi}{2}\right)\)
\(\,\,\,\,\,\)The critical numbers are \(0,\frac{\pi}{2},\frac{3\pi}{2},2\pi\).
\(\textbf{15)}\) \( f(x)=\cos{x}\text{ on }[0,2\pi] \)
The critical numbers are \(0,\pi,2\pi\)
The increasing interval is \((\pi,2\pi)\)
The decreasing interval is \((0,\pi)\)
\(\,\,\,\,\,\,f(x)=\cos{x}\)
\(\,\,\,\,\,\,f'(x)=-\sin{x}\)
\(\,\,\,\,\,\,-\sin{x}=0\)
\(\,\,\,\,\,\,x=0,\pi,2\pi\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(0,\pi)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(\pi,2\pi)\)
\(\,\,\,\,\,\)The critical numbers are \(0,\pi,2\pi\).
\(\,\,\,\,\,\,f(x)=\cos{x}\)
\(\,\,\,\,\,\,f'(x)=-\sin{x}\)
\(\,\,\,\,\,\,-\sin{x}=0\)
\(\,\,\,\,\,\,x=0,\pi,2\pi\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(0,\pi)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(\pi,2\pi)\)
\(\,\,\,\,\,\)The critical numbers are \(0,\pi,2\pi\).
\(\textbf{16)}\) \( f(x)=x^3-12x+1 \)
The critical numbers are \(-2\) and \(2\)
The increasing intervals are \((-\infty,-2)\) and \((2,\infty)\)
The decreasing interval is \((-2,2)\)
\(\,\,\,\,\,\,f(x)=x^3-12x+1\)
\(\,\,\,\,\,\,f'(x)=3x^2-12\)
\(\,\,\,\,\,\,3x^2-12=0\)
\(\,\,\,\,\,\,x^2=4\)
\(\,\,\,\,\,\,x=-2\text{ or }x=2\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,-2)\cup(2,\infty)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-2,2)\)
\(\,\,\,\,\,\)The critical numbers are \(-2\) and \(2\).
\(\,\,\,\,\,\,f(x)=x^3-12x+1\)
\(\,\,\,\,\,\,f'(x)=3x^2-12\)
\(\,\,\,\,\,\,3x^2-12=0\)
\(\,\,\,\,\,\,x^2=4\)
\(\,\,\,\,\,\,x=-2\text{ or }x=2\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,-2)\cup(2,\infty)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-2,2)\)
\(\,\,\,\,\,\)The critical numbers are \(-2\) and \(2\).
\(\textbf{17)}\) \( f(x)=x^5-5x \)
The critical numbers are \(-1\) and \(1\)
The increasing intervals are \((-\infty,-1)\) and \((1,\infty)\)
The decreasing interval is \((-1,1)\)
\(\,\,\,\,\,\,f(x)=x^5-5x\)
\(\,\,\,\,\,\,f'(x)=5x^4-5\)
\(\,\,\,\,\,\,5x^4-5=0\)
\(\,\,\,\,\,\,x^4=1\)
\(\,\,\,\,\,\,x=-1\text{ or }x=1\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,-1)\cup(1,\infty)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-1,1)\)
\(\,\,\,\,\,\)The critical numbers are \(-1\) and \(1\).
\(\,\,\,\,\,\,f(x)=x^5-5x\)
\(\,\,\,\,\,\,f'(x)=5x^4-5\)
\(\,\,\,\,\,\,5x^4-5=0\)
\(\,\,\,\,\,\,x^4=1\)
\(\,\,\,\,\,\,x=-1\text{ or }x=1\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,-1)\cup(1,\infty)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-1,1)\)
\(\,\,\,\,\,\)The critical numbers are \(-1\) and \(1\).
\(\textbf{18)}\) \( f(x)=\frac{x^2}{x+1} \)
The critical number is \(0\). \(x=-1\) is not in the domain.
The increasing intervals are \((-\infty,-1)\) and \((0,\infty)\)
The decreasing interval is \((-1,0)\)
\(\,\,\,\,\,\,f(x)=\frac{x^2}{x+1}\)
\(\,\,\,\,\,\,f'(x)=\frac{(x+1)(2x)-x^2(1)}{(x+1)^2}\)
\(\,\,\,\,\,\,f'(x)=\frac{2x^2+2x-x^2}{(x+1)^2}\)
\(\,\,\,\,\,\,f'(x)=\frac{x^2+2x}{(x+1)^2}\)
\(\,\,\,\,\,\,f'(x)=\frac{x(x+2)}{(x+1)^2}\)
\(\,\,\,\,\,\,x=0\text{ is a critical number.}\)
\(\,\,\,\,\,\,x=-1\text{ is not in the domain.}\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,-1)\cup(0,\infty)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-1,0)\)
\(\,\,\,\,\,\)The critical number is \(0\).
\(\,\,\,\,\,\,f(x)=\frac{x^2}{x+1}\)
\(\,\,\,\,\,\,f'(x)=\frac{(x+1)(2x)-x^2(1)}{(x+1)^2}\)
\(\,\,\,\,\,\,f'(x)=\frac{2x^2+2x-x^2}{(x+1)^2}\)
\(\,\,\,\,\,\,f'(x)=\frac{x^2+2x}{(x+1)^2}\)
\(\,\,\,\,\,\,f'(x)=\frac{x(x+2)}{(x+1)^2}\)
\(\,\,\,\,\,\,x=0\text{ is a critical number.}\)
\(\,\,\,\,\,\,x=-1\text{ is not in the domain.}\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }(-\infty,-1)\cup(0,\infty)\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }(-1,0)\)
\(\,\,\,\,\,\)The critical number is \(0\).
\(\textbf{19)}\) \( f(x)=x^2\ln{x} \)
The critical number is \(e^{-1/2}\)
The increasing interval is \(\left(e^{-1/2},\infty\right)\)
The decreasing interval is \(\left(0,e^{-1/2}\right)\)
\(\,\,\,\,\,\,f(x)=x^2\ln{x}\)
\(\,\,\,\,\,\,f'(x)=2x\ln{x}+x^2\cdot\frac{1}{x}\)
\(\,\,\,\,\,\,f'(x)=2x\ln{x}+x\)
\(\,\,\,\,\,\,f'(x)=x(2\ln{x}+1)\)
\(\,\,\,\,\,\,x(2\ln{x}+1)=0\)
\(\,\,\,\,\,\,2\ln{x}+1=0\)
\(\,\,\,\,\,\,\ln{x}=-\frac{1}{2}\)
\(\,\,\,\,\,\,x=e^{-1/2}\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }\left(0,e^{-1/2}\right)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }\left(e^{-1/2},\infty\right)\)
\(\,\,\,\,\,\)The critical number is \(e^{-1/2}\).
\(\,\,\,\,\,\,f(x)=x^2\ln{x}\)
\(\,\,\,\,\,\,f'(x)=2x\ln{x}+x^2\cdot\frac{1}{x}\)
\(\,\,\,\,\,\,f'(x)=2x\ln{x}+x\)
\(\,\,\,\,\,\,f'(x)=x(2\ln{x}+1)\)
\(\,\,\,\,\,\,x(2\ln{x}+1)=0\)
\(\,\,\,\,\,\,2\ln{x}+1=0\)
\(\,\,\,\,\,\,\ln{x}=-\frac{1}{2}\)
\(\,\,\,\,\,\,x=e^{-1/2}\)
\(\,\,\,\,\,\,f'(x) \lt 0\text{ on }\left(0,e^{-1/2}\right)\)
\(\,\,\,\,\,\,f'(x) \gt 0\text{ on }\left(e^{-1/2},\infty\right)\)
\(\,\,\,\,\,\)The critical number is \(e^{-1/2}\).
\(\textbf{20)}\) \( f(x)=\arctan{x} \)
There are no critical numbers.
The increasing interval is \((-\infty,\infty)\)
The decreasing interval is \(\emptyset\)
\(\,\,\,\,\,\,f(x)=\arctan{x}\)
\(\,\,\,\,\,\,f'(x)=\frac{1}{1+x^2}\)
\(\,\,\,\,\,\,\frac{1}{1+x^2}>0\text{ for all }x\)
\(\,\,\,\,\,\,\text{The derivative is never }0\text{ or undefined.}\)
\(\,\,\,\,\,\)There are no critical numbers. The function is increasing on \((-\infty,\infty)\).
\(\,\,\,\,\,\,f(x)=\arctan{x}\)
\(\,\,\,\,\,\,f'(x)=\frac{1}{1+x^2}\)
\(\,\,\,\,\,\,\frac{1}{1+x^2}>0\text{ for all }x\)
\(\,\,\,\,\,\,\text{The derivative is never }0\text{ or undefined.}\)
\(\,\,\,\,\,\)There are no critical numbers. The function is increasing on \((-\infty,\infty)\).
See Related Pages\(\)
\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)
\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)
\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)
\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)
\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)
\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)
\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)
\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)
\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)
\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)
\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)
\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)