suppose f(x) is a continuous function with a continuous derivative, with domain -∞ < x < ∞. a portion of the…

suppose f(x) is a continuous function with a continuous derivative, with domain -∞ < 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 minimum value at x = 3 the derivative of f(x) has a maximum value at x = dne 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 =

suppose f(x) is a continuous function with a continuous derivative, with domain -∞ < 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 minimum value at x = 3 the derivative of f(x) has a maximum value at x = dne 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 $f'(x)$ from the graph

From the graph of $y = f'(x)$, we see that $f'(x)$ is negative for $x\in(1,4)$ and positive for $x < 1$ and $x>4$. So $f(x)$ has a local maximum at $x = 1$ (since $f'(x)$ changes from positive to negative at $x = 1$) and a local minimum at $x = 4$ (since $f'(x)$ changes from negative to positive at $x = 4$).

Answer:

The function $f(x)$ has a local maximum value at $x = 1$. The function $f(x)$ has a local minimum value at $x = 4$.