Question

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

Answer 1

Please see below.

Explanation:

For the graph of a function, `f(x)`

Find critical numbers for `f`. These are the values in the domain of `f` at which `f'(x) = 0` or `f'(x)` 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'(x)` changes from negative to positive as x values move left to right past `c`, then `f(c)` is a local minimum for `f`.

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