Question

he graph of f is given.
(a) on what open intervals is f increasing?
(b) on what open intervals is f decreasing?
(c) what are the x - values of the local extrema?
(d) on what open intervals is f concave up?
(e) on what open intervals is f concave down?
(f) what are the x - values of the inflection points?
(c) determine the x - values corresponding to local minima. select the correct choice and,
if necessary, fill in the answer box to complete your choice.
a. the graph of f has one or more local minima at x=
(round to the nearest integer as needed. use a comma to separate answers as needed )
b. the graph of f has no local minima.

Explanation:

Step1: Recall the first - derivative test

A function \(y = f(x)\) has a local minimum at a point \(x = c\) if \(f^{\prime}(x)\) changes sign from negative to positive at \(x = c\).

Step2: Analyze the sign change of \(f^{\prime}(x)\)

Looking at the graph of \(y = f^{\prime}(x)\), we observe the sign change of \(f^{\prime}(x)\).
We can see that \(f^{\prime}(x)\) changes from negative to positive at \(x=- 10\) (approximate value by looking at the graph where the \(x\) - axis crossing from below to above).

Answer:

A. The graph of \(f\) has one or more local minima at \(x=-10\)