sketch the graph of a function f that has a local minimum value where f(b)=0. choose the correct graph below.
sketch the graph of a function f that has a local minimum value where f(b)=0. choose the correct graph below.
Answer
Explanation:
Step1: Recall derivative - minimum relationship
At a local minimum of a differentiable function, the derivative is zero and the function changes from decreasing to increasing.
Step2: Analyze each option
In option A, at (x = b) the function has a local maximum since it changes from increasing to decreasing. In option B, at (x = b) the function has a local maximum. In option C, at (x = b) the function has a local maximum. In option D, at (x = b), (f^{\prime}(b)=0) and the function changes from decreasing to increasing, so it has a local minimum at (x = b).
Answer:
D.