# Analyzing the second derivative to find inflection points

Learn how the second derivative of a function is used in order to find the function's inflection points. Learn which common mistakes to avoid in the process.

We can find the inflection points of a function by analyzing its second derivative.

## Example: Finding the inflection points of $f(x)=x^5+\dfrac53x^4$

**Step 1: Finding the second derivative**

To find the inflection points of $f$, we need to use $f''$:

**Step 2: Finding all candidates**

Similar to critical points, these are points where $f''(x)=0$ or where $f''(x)$ is undefined.

$f''$ is zero at $x=0$ and $x=-1$, and it's defined for all real numbers. So $x=0$ and $x=-1$ are our candidates.

**Step 3: Analyzing concavity**

Interval | Test $x$-value | $f''(x)$ | Conclusion |
---|---|---|---|

$x<-1$ | $x=-2$ | $f''(-2)=-80<0$ | $f$ is concave down $\cap$ |

$-1<x<0$ | $x=-0.5$ | $f''(-0.5)=2.5>0$ | $f$ is concave up $\cup$ |

$x>0$ | $x=1$ | $f''(1)=40>0$ | $f$ is concave up $\cup$ |

**Step 4: Finding inflection points**

Now that we know the intervals where $f$ is concave up or down, we can find its inflection points (i.e. where the concavity changes direction).

- $f$ is concave down before $x=-1$, concave up after it, and is defined at $x=-1$. So $f$ has an inflection point at $x=-1$.
- $f$ is concave up before
*and after*$x=0$, so it doesn't have an inflection point there.

We can verify our result by looking at the graph of $f$.

### Common mistake: not checking the candidates

**Remember:**We must not assume that any point where $f''(x)=0$ (or where $f''(x)$ is undefined) is an inflection point. Instead, we should check our candidates to see if the second derivative changes signs at those points and the function is defined at those points.

### Common mistake: not including points where the derivative is undefined

**Remember:**Our candidates for inflection points are points where the second derivative is equal to zero

*points where the second derivative is undefined. Ignoring points where the second derivative is undefined will often result in a wrong answer.*

**and**### Common mistake: looking at the first derivative instead of the second derivative

**Remember**: When looking for inflection points, we must always analyze where the

*second*derivative changes its sign. Doing this for the first derivative will give us relative extremum points, not inflection points.

*Want more practice? Try this exercise.*