Concavity describes whether a graph bends upward like a cup or downward like an upside-down cup. The second derivative tells us concavity: when \(f”(x) \gt 0\), the graph is concave up, and when \(f”(x) \lt 0\), the graph is concave down. Inflection points occur where the graph changes concavity, usually at values where \(f”(x)=0\) or where \(f”(x)\) is undefined.
Notes
Practice Problems
For each function
a. Find the points of inflection
b. Find the open intervals where f is concave up
c. Find the open intervals where f is concave down
\(\textbf{1)}\) \( f(x)=2x^2+4x+3 \)
point of inflection: none
concave up \((-\infty,\infty) \)
concave down \( \emptyset \)
\(\,\,\,\,\,\,f(x)=2x^2+4x+3\)
\(\,\,\,\,\,\,f'(x)=4x+4\)
\(\,\,\,\,\,\,f”(x)=4\)
\(\,\,\,\,\,\,4>0\text{ for all }x\)
\(\,\,\,\,\,\,\text{The graph is concave up on }(-\infty,\infty).\)
\(\,\,\,\,\,\,\text{Since the concavity never changes, there is no point of inflection.}\)
\(\textbf{2)}\) \( f(x)=\frac{1}{5}x^5-16x+5 \)
point of inflection: \( (0,5) \)
concave up \( (0, \infty) \)
concave down \( (- \infty,0) \)
\(\,\,\,\,\,\,f(x)=\frac{1}{5}x^5-16x+5\)
\(\,\,\,\,\,\,f'(x)=x^4-16\)
\(\,\,\,\,\,\,f”(x)=4x^3\)
\(\,\,\,\,\,\,4x^3=0\)
\(\,\,\,\,\,\,x=0\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }(-\infty,0)\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }(0,\infty)\)
\(\,\,\,\,\,\,f(0)=5\)
\(\,\,\,\,\,\,\text{The point of inflection is }(0,5).\)
\(\textbf{3)}\) \( f(x)=-3x+2 \)
point of inflection: none
concave up \( \emptyset \)
concave down \( \emptyset \)
\(\,\,\,\,\,\,f(x)=-3x+2\)
\(\,\,\,\,\,\,f'(x)=-3\)
\(\,\,\,\,\,\,f”(x)=0\)
\(\,\,\,\,\,\,\text{The second derivative is never positive or negative.}\)
\(\,\,\,\,\,\,\text{The graph is a line, so it is not concave up or concave down.}\)
\(\,\,\,\,\,\,\text{There is no point of inflection.}\)
\(\textbf{4)}\) \( f(x)=4x^3+1 \)
point of inflection: \( (0,1) \) (plateau)
concave up \( (0,\infty) \)
concave down \( (-\infty,0) \)
\(\,\,\,\,\,\,f(x)=4x^3+1\)
\(\,\,\,\,\,\,f'(x)=12x^2\)
\(\,\,\,\,\,\,f”(x)=24x\)
\(\,\,\,\,\,\,24x=0\)
\(\,\,\,\,\,\,x=0\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }(-\infty,0)\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }(0,\infty)\)
\(\,\,\,\,\,\,f(0)=1\)
\(\,\,\,\,\,\,\text{The point of inflection is }(0,1).\)
\(\textbf{5)}\) \( f(x)=x^3-3x^2 \)
point of inflection: \((1,-2)\)
concave up \((1,\infty)\)
concave down \((-\infty,1)\)
\(\,\,\,\,\,\,f(x)=x^3-3x^2\)
\(\,\,\,\,\,\,f'(x)=3x^2-6x\)
\(\,\,\,\,\,\,f”(x)=6x-6\)
\(\,\,\,\,\,\,6x-6=0\)
\(\,\,\,\,\,\,x=1\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }(-\infty,1)\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }(1,\infty)\)
\(\,\,\,\,\,\,f(1)=1-3=-2\)
\(\,\,\,\,\,\,\text{The point of inflection is }(1,-2).\)
\(\textbf{6)}\) \( f(x)=x^4-6x^2 \)
points of inflection: \((-1,-5)\) and \((1,-5)\)
concave up \((-\infty,-1)\cup(1,\infty)\)
concave down \((-1,1)\)
\(\,\,\,\,\,\,f(x)=x^4-6x^2\)
\(\,\,\,\,\,\,f'(x)=4x^3-12x\)
\(\,\,\,\,\,\,f”(x)=12x^2-12\)
\(\,\,\,\,\,\,12x^2-12=0\)
\(\,\,\,\,\,\,x^2=1\)
\(\,\,\,\,\,\,x=-1\text{ or }x=1\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }(-\infty,-1)\cup(1,\infty)\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }(-1,1)\)
\(\,\,\,\,\,\,f(-1)=-5\text{ and }f(1)=-5\)
\(\,\,\,\,\,\,\text{The points of inflection are }(-1,-5)\text{ and }(1,-5).\)
\(\textbf{7)}\) \( f(x)=-x^4+8x^2 \)
points of inflection: \(\left(-\frac{2\sqrt{3}}{3},\frac{80}{9}\right)\) and \(\left(\frac{2\sqrt{3}}{3},\frac{80}{9}\right)\)
concave up \(\left(-\frac{2\sqrt{3}}{3},\frac{2\sqrt{3}}{3}\right)\)
concave down \(\left(-\infty,-\frac{2\sqrt{3}}{3}\right)\cup\left(\frac{2\sqrt{3}}{3},\infty\right)\)
\(\,\,\,\,\,\,f(x)=-x^4+8x^2\)
\(\,\,\,\,\,\,f'(x)=-4x^3+16x\)
\(\,\,\,\,\,\,f”(x)=-12x^2+16\)
\(\,\,\,\,\,\,-12x^2+16=0\)
\(\,\,\,\,\,\,x^2=\frac{4}{3}\)
\(\,\,\,\,\,\,x=\pm\frac{2\sqrt{3}}{3}\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ between the two critical values.}\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ outside the two critical values.}\)
\(\,\,\,\,\,\,f\left(\pm\frac{2\sqrt{3}}{3}\right)=\frac{80}{9}\)
\(\,\,\,\,\,\,\text{The points of inflection are }\left(-\frac{2\sqrt{3}}{3},\frac{80}{9}\right)\text{ and }\left(\frac{2\sqrt{3}}{3},\frac{80}{9}\right).\)
\(\textbf{8)}\) \( f(x)=x^4-8x^3 \)
points of inflection: \((0,0)\) and \((4,-256)\)
concave up \((-\infty,0)\cup(4,\infty)\)
concave down \((0,4)\)
\(\,\,\,\,\,\,f(x)=x^4-8x^3\)
\(\,\,\,\,\,\,f'(x)=4x^3-24x^2\)
\(\,\,\,\,\,\,f”(x)=12x^2-48x\)
\(\,\,\,\,\,\,12x^2-48x=0\)
\(\,\,\,\,\,\,12x(x-4)=0\)
\(\,\,\,\,\,\,x=0\text{ or }x=4\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }(-\infty,0)\cup(4,\infty)\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }(0,4)\)
\(\,\,\,\,\,\,f(0)=0\text{ and }f(4)=-256\)
\(\,\,\,\,\,\,\text{The points of inflection are }(0,0)\text{ and }(4,-256).\)
\(\textbf{9)}\) \( f(x)=x^5-10x^3 \)
points of inflection: \((-\sqrt{3},21\sqrt{3})\), \((0,0)\), and \((\sqrt{3},-21\sqrt{3})\)
concave up \((-\sqrt{3},0)\cup(\sqrt{3},\infty)\)
concave down \((-\infty,-\sqrt{3})\cup(0,\sqrt{3})\)
\(\,\,\,\,\,\,f(x)=x^5-10x^3\)
\(\,\,\,\,\,\,f'(x)=5x^4-30x^2\)
\(\,\,\,\,\,\,f”(x)=20x^3-60x\)
\(\,\,\,\,\,\,20x^3-60x=0\)
\(\,\,\,\,\,\,20x(x^2-3)=0\)
\(\,\,\,\,\,\,x=-\sqrt{3},\,0,\,\sqrt{3}\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }(-\infty,-\sqrt{3})\cup(0,\sqrt{3})\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }(-\sqrt{3},0)\cup(\sqrt{3},\infty)\)
\(\,\,\,\,\,\,f(-\sqrt{3})=21\sqrt{3},\quad f(0)=0,\quad f(\sqrt{3})=-21\sqrt{3}\)
\(\,\,\,\,\,\,\text{The points of inflection are }(-\sqrt{3},21\sqrt{3}),\,(0,0),\text{ and }(\sqrt{3},-21\sqrt{3}).\)
\(\textbf{10)}\) \( f(x)=e^x \)
point of inflection: none
concave up \((-\infty,\infty)\)
concave down \(\emptyset\)
\(\,\,\,\,\,\,f(x)=e^x\)
\(\,\,\,\,\,\,f'(x)=e^x\)
\(\,\,\,\,\,\,f”(x)=e^x\)
\(\,\,\,\,\,\,e^x \gt 0\text{ for all }x\)
\(\,\,\,\,\,\,\text{The graph is concave up on }(-\infty,\infty).\)
\(\,\,\,\,\,\,\text{Since the concavity never changes, there is no point of inflection.}\)
\(\textbf{11)}\) \( f(x)=xe^x \)
point of inflection: \(\left(-2,-\frac{2}{e^2}\right)\)
concave up \((-2,\infty)\)
concave down \((-\infty,-2)\)
\(\,\,\,\,\,\,f(x)=xe^x\)
\(\,\,\,\,\,\,f'(x)=e^x+xe^x=e^x(x+1)\)
\(\,\,\,\,\,\,f”(x)=e^x(x+1)+e^x=e^x(x+2)\)
\(\,\,\,\,\,\,e^x(x+2)=0\)
\(\,\,\,\,\,\,x=-2\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }(-\infty,-2)\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }(-2,\infty)\)
\(\,\,\,\,\,\,f(-2)=-2e^{-2}=-\frac{2}{e^2}\)
\(\,\,\,\,\,\,\text{The point of inflection is }\left(-2,-\frac{2}{e^2}\right).\)
\(\textbf{12)}\) \( f(x)=\ln{x} \)
point of inflection: none
concave up \(\emptyset\)
concave down \((0,\infty)\)
\(\,\,\,\,\,\,f(x)=\ln{x}\)
\(\,\,\,\,\,\,f'(x)=\frac{1}{x}\)
\(\,\,\,\,\,\,f”(x)=-\frac{1}{x^2}\)
\(\,\,\,\,\,\,-\frac{1}{x^2}<0\text{ for all }x>0\)
\(\,\,\,\,\,\,\text{The graph is concave down on }(0,\infty).\)
\(\,\,\,\,\,\,\text{Since the concavity never changes, there is no point of inflection.}\)
\(\textbf{13)}\) \( f(x)=\sin{x}\text{ on }[0,2\pi] \)
point of inflection: \((\pi,0)\)
concave up \((\pi,2\pi)\)
concave down \((0,\pi)\)
\(\,\,\,\,\,\,f(x)=\sin{x}\)
\(\,\,\,\,\,\,f'(x)=\cos{x}\)
\(\,\,\,\,\,\,f”(x)=-\sin{x}\)
\(\,\,\,\,\,\,-\sin{x}=0\)
\(\,\,\,\,\,\,x=0,\pi,2\pi\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }(0,\pi)\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }(\pi,2\pi)\)
\(\,\,\,\,\,\,f(\pi)=0\)
\(\,\,\,\,\,\,\text{The point of inflection is }(\pi,0).\)
\(\textbf{14)}\) \( f(x)=\cos{x}\text{ on }[0,2\pi] \)
points of inflection: \(\left(\frac{\pi}{2},0\right)\) and \(\left(\frac{3\pi}{2},0\right)\)
concave up \(\left(\frac{\pi}{2},\frac{3\pi}{2}\right)\)
concave down \(\left(0,\frac{\pi}{2}\right)\cup\left(\frac{3\pi}{2},2\pi\right)\)
\(\,\,\,\,\,\,f(x)=\cos{x}\)
\(\,\,\,\,\,\,f'(x)=-\sin{x}\)
\(\,\,\,\,\,\,f”(x)=-\cos{x}\)
\(\,\,\,\,\,\,-\cos{x}=0\)
\(\,\,\,\,\,\,x=\frac{\pi}{2},\frac{3\pi}{2}\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }\left(0,\frac{\pi}{2}\right)\cup\left(\frac{3\pi}{2},2\pi\right)\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }\left(\frac{\pi}{2},\frac{3\pi}{2}\right)\)
\(\,\,\,\,\,\,f\left(\frac{\pi}{2}\right)=0\text{ and }f\left(\frac{3\pi}{2}\right)=0\)
\(\,\,\,\,\,\,\text{The points of inflection are }\left(\frac{\pi}{2},0\right)\text{ and }\left(\frac{3\pi}{2},0\right).\)
\(\textbf{15)}\) \( f(x)=\arctan{x} \)
point of inflection: \((0,0)\)
concave up \((-\infty,0)\)
concave down \((0,\infty)\)
\(\,\,\,\,\,\,f(x)=\arctan{x}\)
\(\,\,\,\,\,\,f'(x)=\frac{1}{1+x^2}\)
\(\,\,\,\,\,\,f”(x)=\frac{-2x}{(1+x^2)^2}\)
\(\,\,\,\,\,\,\frac{-2x}{(1+x^2)^2}=0\)
\(\,\,\,\,\,\,x=0\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }(-\infty,0)\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }(0,\infty)\)
\(\,\,\,\,\,\,f(0)=0\)
\(\,\,\,\,\,\,\text{The point of inflection is }(0,0).\)
\(\textbf{16)}\) \( f(x)=\frac{1}{x} \)
point of inflection: none
concave up \((0,\infty)\)
concave down \((-\infty,0)\)
\(\,\,\,\,\,\,f(x)=\frac{1}{x}\)
\(\,\,\,\,\,\,f'(x)=-\frac{1}{x^2}\)
\(\,\,\,\,\,\,f”(x)=\frac{2}{x^3}\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }(-\infty,0)\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }(0,\infty)\)
\(\,\,\,\,\,\,x=0\text{ is not in the domain of }f(x).\)
\(\,\,\,\,\,\,\text{Since an inflection point must be on the graph, there is no point of inflection.}\)
\(\textbf{17)}\) \( f(x)=(x-1)^3+2 \)
point of inflection: \((1,2)\)
concave up \((1,\infty)\)
concave down \((-\infty,1)\)
\(\,\,\,\,\,\,f(x)=(x-1)^3+2\)
\(\,\,\,\,\,\,f'(x)=3(x-1)^2\)
\(\,\,\,\,\,\,f”(x)=6(x-1)\)
\(\,\,\,\,\,\,6(x-1)=0\)
\(\,\,\,\,\,\,x=1\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }(-\infty,1)\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }(1,\infty)\)
\(\,\,\,\,\,\,f(1)=2\)
\(\,\,\,\,\,\,\text{The point of inflection is }(1,2).\)
\(\textbf{18)}\) \( f(x)=x^6-5x^4 \)
points of inflection: \((-\sqrt{2},-12)\) and \((\sqrt{2},-12)\)
concave up \((-\infty,-\sqrt{2})\cup(\sqrt{2},\infty)\)
concave down \((-\sqrt{2},0)\cup(0,\sqrt{2})\)
\(\,\,\,\,\,\,f(x)=x^6-5x^4\)
\(\,\,\,\,\,\,f'(x)=6x^5-20x^3\)
\(\,\,\,\,\,\,f”(x)=30x^4-60x^2\)
\(\,\,\,\,\,\,30x^4-60x^2=0\)
\(\,\,\,\,\,\,30x^2(x^2-2)=0\)
\(\,\,\,\,\,\,x=0,\,-\sqrt{2},\,\sqrt{2}\)
\(\,\,\,\,\,\,\text{The second derivative does not change sign at }x=0.\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }(-\infty,-\sqrt{2})\cup(\sqrt{2},\infty)\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }(-\sqrt{2},0)\cup(0,\sqrt{2})\)
\(\,\,\,\,\,\,f(-\sqrt{2})=-12\text{ and }f(\sqrt{2})=-12\)
\(\,\,\,\,\,\,\text{The points of inflection are }(-\sqrt{2},-12)\text{ and }(\sqrt{2},-12).\)
\(\textbf{19)}\) \( f(x)=x^4+x^3 \)
points of inflection: \(\left(-\frac{1}{2},-\frac{1}{16}\right)\) and \((0,0)\)
concave up \(\left(-\infty,-\frac{1}{2}\right)\cup(0,\infty)\)
concave down \(\left(-\frac{1}{2},0\right)\)
\(\,\,\,\,\,\,f(x)=x^4+x^3\)
\(\,\,\,\,\,\,f'(x)=4x^3+3x^2\)
\(\,\,\,\,\,\,f”(x)=12x^2+6x\)
\(\,\,\,\,\,\,12x^2+6x=0\)
\(\,\,\,\,\,\,6x(2x+1)=0\)
\(\,\,\,\,\,\,x=-\frac{1}{2}\text{ or }x=0\)
\(\,\,\,\,\,\,f”(x) \gt 0\text{ on }\left(-\infty,-\frac{1}{2}\right)\cup(0,\infty)\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ on }\left(-\frac{1}{2},0\right)\)
\(\,\,\,\,\,\,f\left(-\frac{1}{2}\right)=-\frac{1}{16}\text{ and }f(0)=0\)
\(\,\,\,\,\,\,\text{The points of inflection are }\left(-\frac{1}{2},-\frac{1}{16}\right)\text{ and }(0,0).\)
\(\textbf{20)}\) \( f(x)=\sqrt{x} \)
point of inflection: none
concave up \(\emptyset\)
concave down \((0,\infty)\)
\(\,\,\,\,\,\,f(x)=\sqrt{x}=x^{1/2}\)
\(\,\,\,\,\,\,f'(x)=\frac{1}{2}x^{-1/2}\)
\(\,\,\,\,\,\,f”(x)=-\frac{1}{4}x^{-3/2}\)
\(\,\,\,\,\,\,f”(x) \lt 0\text{ for all }x>0\)
\(\,\,\,\,\,\,\text{The graph is concave down on }(0,\infty).\)
\(\,\,\,\,\,\,\text{Since the concavity never changes, there is no point of inflection.}\)
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