a portion of the graph of y = f(x) is shown. remember this is the graph of y = f(x) and not the graph of y =…

a portion of the graph of y = f(x) is shown. remember this is the graph of y = f(x) and not the graph of y = f(x). for all of the following, enter the x - value where the extrema occurs, or dne if no such x - value exists. the following questions are about the derivative of f(x). based on the graph: the derivative of f(x) has a maximum value at x = dne the derivative of f(x) has a minimum value at x = 6 the following questions are about the function f(x). based on the graph: the function f(x) has a local maximum value at x = the function f(x) has a local minimum value at x =

a portion of the graph of y = f(x) is shown. remember this is the graph of y = f(x) and not the graph of y = f(x). for all of the following, enter the x - value where the extrema occurs, or dne if no such x - value exists. the following questions are about the derivative of f(x). based on the graph: the derivative of f(x) has a maximum value at x = dne the derivative of f(x) has a minimum value at x = 6 the following questions are about the function f(x). based on the graph: the function f(x) has a local maximum value at x = the function f(x) has a local minimum value at x =

Answer

Explanation:

Step1: Recall the relationship between $f(x)$ and $f'(x)$

A local maximum of $f(x)$ occurs where $f'(x)$ changes sign from positive to negative. A local minimum of $f(x)$ occurs where $f'(x)$ changes sign from negative to positive.

Step2: Analyze the sign - change of $f'(x)$

Looking at the graph of $y = f'(x)$, we see that $f'(x)$ is negative for $x<6$ and positive for $x > 6$.

Step3: Determine local extrema of $f(x)$

Since $f'(x)$ changes sign from negative to positive at $x = 6$, the function $f(x)$ has a local minimum at $x=6$. There is no point where $f'(x)$ changes sign from positive to negative in the given graph, so there is no local maximum.

Answer:

The function $f(x)$ has a local maximum value at $x=$ DNE The function $f(x)$ has a local minimum value at $x = 6$