Question

current attempt in progress
the figure below is the graph of a derivative f. give the x - values of the critical points of f. at which critical points does f have local maxima, local minima or neither?
enter the critical points in increasing order. round your answers to the nearest integers.
if there are less than three critical points, enter the critical points first, then enter na in the remaining answer field(s) and select
either a maximum nor a minimum\ from the dropdown menu.

Explanation:

Step1: Recall critical - point definition

Critical points of $f$ occur where $f'(x)=0$ or $f'(x)$ is undefined. From the graph of $f'$, $f'(x) = 0$ at the $x$ - values where the graph of $f'$ crosses the $x$ - axis.

Step2: Identify $x$ - values of $f'(x)=0$

The graph of $f'$ crosses the $x$ - axis at $x=-2,x = - 1,x = 1$.

Step3: Use the first - derivative test

If $f'$ changes sign from positive to negative at a critical point $c$, then $f(c)$ is a local maximum. If $f'$ changes sign from negative to positive at $c$, then $f(c)$ is a local minimum. If $f'$ does not change sign at $c$, then $f(c)$ is neither a local maximum nor a local minimum.

• For $x=-2$: $f'$ changes sign from positive to negative, so $f$ has a local maximum at $x=-2$.
• For $x=-1$: $f'$ changes sign from negative to positive, so $f$ has a local minimum at $x=-1$.
• For $x = 1$: $f'$ changes sign from negative to positive, so $f$ has a local minimum at $x = 1$.

Answer:

$x=-2$ is a local maximum
$x=-1$ is a local minimum
$x = 1$ is a local minimum