# Are the inflection points where f'(x) = zero or where the graph changes from concave up to concave down?

May 23, 2015

The inflection point is a point where the graph of the function changes from concave up to concave down or vice versa.

To calculate these points you have to find places where $f ' ' \left(x\right) = 0$ and check if the second derivative changes sign at this point.

For example to find the points of inflection for $f \left(x\right) = {x}^{7}$you have to calculate $f ' ' \left(x\right)$ first.

$f ' \left(x\right) = 7 {x}^{6}$
$f ' ' \left(x\right) = 42 {x}^{5}$

Now we have to check where $f ' ' \left(x\right) = 0$
$42 {x}^{5} = 0 \iff x = 0$.

We found that $x = 0$ may be a point of inflection.
To find if it is such point we have to check if $f ' ' \left(x\right)$ changes sign at 0.

To find this we can graph the function:

graph{42x^5 [-3.894, 3.897, -1.95, 1.948]}

We can see that the $f ' ' \left(x\right)$ changes sign at zero, so zero is the point of inflection.

Note
It is important to check to see whether concavity actually changes.

$g \left(x\right) = {x}^{4} + 3 x - 8$

$g ' \left(x\right) + 4 {x}^{3} + 3$

$g ' ' \left(x\right) = 12 {x}^{2}$

Now we have to check where $g ' ' \left(x\right) = 0$
$12 {x}^{2} = 0 \iff x = 0$.

We found that $x = 0$ may be a point of inflection.
To find if it is such point we have to check if $g ' ' \left(x\right)$ changes sign at $0$.

But $g ' ' \left(x\right) = 12 {x}^{2}$ is never negative, it is always positive or $0$, therefors the sign of $g ' ' \left(x\right)$ does not change, so there are no inflection points.

May 23, 2015

I have been taught and, following our textbook's lead, I continue to teach , that an inflection point is a point on the graph at which the concavity changes.

#### Explanation:

Using this terminology:
For $f \left(x\right) = {x}^{7}$, the inflection point is $\left(0 , 0\right)$

The function: $h \left(x\right) = \frac{1}{x}$ is concave down on $\left(- \infty , 0\right)$ and concave up on $\left(0 , \infty\right)$.
The concavity is not the same on the entire graph, but there is no inflection point, because there is no point on the graph at $x = 0$.