Concavity and inflection points

Where $f''(x)>0$ the graph is concave up, and where $f''(x)<0$ it is concave down. $f''(x)$ measures how the slope $f'(x)$ changes, so the graph is concave up where the slope keeps increasing.

An inflection point is where the concavity switches. Solve $f''(x)=0$, then check whether $f''(x)$ changes sign on either side.

Here the sign changes at $x=4$, so $x=4$ is an inflection point.

Examples

1. 01A function has this second derivative. What is the x-coordinate of its inflection point?$\displaystyle f''(x)=6x+6$AnswerHide

   Answer$\displaystyle -1$
2. 02For which x is f''(x) positive?$\displaystyle f''(x)=-6x+30$AnswerHide

   Answer$\displaystyle x<5$
3. 03Which graph is concave up everywhere?$\displaystyle$AnswerHide

   Answer$\displaystyle y=5x^2$
4. 04If this is f''(x), on which interval is the graph of f(x) concave up?$\displaystyle f''(x)=6x-6$AnswerHide

   Answer$\displaystyle x>1$
5. 05A function has this second derivative. What is the x-coordinate of its inflection point?$\displaystyle f''(x)=6x+36$AnswerHide

   Answer$\displaystyle -6$
6. 06For which x is f''(x) positive?$\displaystyle f''(x)=-6x+6$AnswerHide

   Answer$\displaystyle x<1$

FAQ

Q1Is every point with $f''(x)=0$ an inflection point?Show answerHide

A

No. If $f''(x)=3x^2$, then $f''(0)=0$, but $f''(x)>0$ on both sides of $x=0$, so the concavity does not change and there is no inflection point. Always check the sign on both sides.

Q2How do I tell concave up from concave down?Show answerHide

A

Concave up bends like a valley, as in $y=x^2$. Concave down bends like a hill, as in $y=-x^2$. $f''(x)>0$ means concave up.

Practice this pattern in the app, 5 problems a day.

Published 2026-10-13 · Updated 2026-10-13