Derivative functions > Points of inflection
1234Points of inflection

## Theory

Whenever a function shows an accelerated or decelerated increase (or decrease) then you should look at the changes in the slope:

• With an accelerated increase the slope is positive and getting bigger: $f\text{'}$ is increasing. The derivative of $f\text{'}$ itself is therefore positive.

• With an accelerated decrease the slope is negative and getting smaller (more negative): $f\text{'}$ is decreasing. The derivative of $f\text{'}$ itself is therefore negative.

• With a decelerated increase the slope remains positive but is getting smaller: $f\text{'}$ is decreasing. The derivative of $f\text{'}$ itself is therefore negative.

• With a decelerated decrease the slope remains negative but is getting bigger (less negative): $f\text{'}$ is increasing. De derivative of $f\text{'}$ itself is therefore positive.

The derivative of $f\text{'}$ is called the second derivative of $f$.
You write the second derivative like this: $f"$ or $\frac{{\text{d}}^{2}y}{\text{d}{x}^{2}}$.

Any point where the slope changes from increasing to decreasing (or vice versa) is called an inflection point of the graph.
You identify inflection points by using the second derivative to look for the extrema of the first derivative.