How do you find a local minimum of a graph using the first derivative?

Mar 9, 2018

Explanation:

For the graph of a function, $f \left(x\right)$

Find critical numbers for $f$. These are the values in the domain of $f$ at which $f ' \left(x\right) = 0$ or $f ' \left(x\right)$ does not exist.

Test each critical number using either the first (or second) derivative test for local extrema.

If $c$ is a critical number for $f$ and if

$f ' \left(x\right)$ changes from negative to positive as x values move left to right past $c$, then $f \left(c\right)$ is a local minimum for $f$.

$f ' \left(x\right)$ changes from positive to negative as x values move left to right past $c$, then $f \left(c\right)$ is a local maximum for $f$.