Increasing and Decreasing Intervals

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

\(\textbf{1)}\) \( f(x)=2x^2+4x+3 \)



 

\(\textbf{2)}\) \( f(x)=\frac{1}{5}x^5-16x+5 \)



 

\(\textbf{3)}\) \( f(x)=-3x+2 \)



 

\(\textbf{4)}\) \( f(x)=4x^3+1 \)



 

\(\textbf{5)}\) \( f(x)=x^3-3x^2 \)



 

\(\textbf{6)}\) \( f(x)=x^4-8x^2 \)



 

\(\textbf{7)}\) \( f(x)=x^4-4x^3 \)



 

\(\textbf{8)}\) \( f(x)=\frac{x+1}{x-2} \)



 

\(\textbf{9)}\) \( f(x)=\sqrt{x+4} \)



 

\(\textbf{10)}\) \( f(x)=\sqrt[3]{x} \)



 

\(\textbf{11)}\) \( f(x)=x+\frac{9}{x} \)



 

\(\textbf{12)}\) \( f(x)=x^2e^{-x} \)



 

\(\textbf{13)}\) \( f(x)=\ln{x}-x \)



 

\(\textbf{14)}\) \( f(x)=\sin{x}\text{ on }[0,2\pi] \)



 

\(\textbf{15)}\) \( f(x)=\cos{x}\text{ on }[0,2\pi] \)



 

\(\textbf{16)}\) \( f(x)=x^3-12x+1 \)



 

\(\textbf{17)}\) \( f(x)=x^5-5x \)



 

\(\textbf{18)}\) \( f(x)=\frac{x^2}{x+1} \)



 

\(\textbf{19)}\) \( f(x)=x^2\ln{x} \)



 

\(\textbf{20)}\) \( f(x)=\arctan{x} \)



 

See Related Pages\(\)

\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)
\(\bullet\text{ Equation of the Tangent Line}\)
\(\,\,\,\,\,\,\,\,f(x)=x^3+3x^2−x \text{ at the point } (2,18)\)
\(\bullet\text{ Derivatives- Constant Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(c)=0\)
\(\bullet\text{ Derivatives- Power Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(x^n)=nx^{n-1}\)
\(\bullet\text{ Derivatives- Constant Multiple Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}(cf(x))=cf'(x)\)
\(\bullet\text{ Derivatives- Sum and Difference Rules}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \pm g(x)]=f'(x) \pm g'(x)\)
\(\bullet\text{ Derivatives- Sin and Cos}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}sin(x)=cos(x)\)
\(\bullet\text{ Derivatives- Product Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(x) \cdot g(x)]=f(x) \cdot g'(x)+f'(x) \cdot g(x)\)
\(\bullet\text{ Derivatives- Quotient Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}\left[\displaystyle\frac{f(x)}{g(x)}\right]=\displaystyle\frac{g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\)
\(\bullet\text{ Derivatives- Chain Rule}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[f(g(x))]= f'(g(x)) \cdot g'(x)\)
\(\bullet\text{ Derivatives- ln(x)}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{d}{dx}[ln(x)]= \displaystyle \frac{1}{x}\)
\(\bullet\text{ Implicit Differentiation}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Horizontal Tangent Line}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Mean Value Theorem}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Related Rates}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Increasing and Decreasing Intervals}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Intervals of concave up and down}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Inflection Points}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
\(\,\,\,\,\,\,\,\,\)Thumbnail of Graph of First and Second Derivatives
\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)

 

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