sketch the graph of a function that has neither a local maximum nor a local minimum at a point where f(x)=…
sketch the graph of a function that has neither a local maximum nor a local minimum at a point where f(x)= 0. choose the correct answer below. a. b. c. d.
Answer
Explanation:
Step1: Recall derivative - extrema relationship
A point $x$ where $f^{\prime}(x) = 0$ is a critical point. For a function to have neither a local max nor a local min at a critical - point, the function must have a horizontal tangent and continue to increase or decrease on both sides of the point.
Step2: Analyze option A
In option A, the function has a local maximum at the point where the tangent is horizontal. So, it is not correct.
Step3: Analyze option B
In option B, the function has a local minimum at the point where the tangent is horizontal. So, it is not correct.
Step4: Analyze option C
This is not a function (it fails the vertical - line test). So, it is not correct.
Step5: Analyze option D
In option D, the function has a horizontal tangent at a point, and the function is increasing on both sides of that point. It has neither a local maximum nor a local minimum at the point where $f^{\prime}(x)=0$.
Answer:
D.