# Critical points & key points

The critical points of a function are at the values of $\displaystyle x$ where

$\displaystyle f{'} ( x ) = 0 \qquad \allowbreak\quad\allowbreak\text{or}\allowbreak\quad\allowbreak \qquad f{'} ( x )$ does not exist.

If the value of $\displaystyle f{'} ( x )$ changes from positive to negative or vice versa across the critical point, then it represents a turning point (local extremum).

The key points of a function are at the values of $\displaystyle x$ where

$\displaystyle f{''} ( x ) = 0 \qquad \allowbreak\quad\allowbreak\text{or}\allowbreak\quad\allowbreak \qquad f{''} ( x )$ does not exist.

If the value of $\displaystyle f{''} ( x )$ changes from positive to negative or vice versa across the key point, then it represents an inflection point.

Critical points can be turning points, and key points can be inflection points, but they aren’t always—just as zeros can indicate that the curve is crossing the x-axis, but not always. All these points are important because they represent *transitions*: from positive to negative, from increasing to decreasing, from concave up to concave down. If you know *when* the three quantities change, you can determine their signs at all points in the domain of the function. You are then well on your way to sketching the curve.