The graphs of the first and second derivatives of a function y = f(x) are given. Select a possible graph of f that passes
through the point P. (NOTE: Vertical scales may vary from graph to graph.)
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Question

Quizplus · Accepted Answer

Since the point P lies on the x-axis, the function must have a critical point at P. This means the derivative must be zero at P. Looking at the graph of the second derivative, we see that it changes sign from negative to positive at P. This means that the function has a local minimum at P. The only graph that satisfies these conditions is D. From the graphs of the first and second derivatives, we can see that the function f(x) has a local maximum at point P. This means that f'(x) changes from positive to negative at point P, and f''(x) is negative at point P. 
Option D is the only graph that satisfies these conditions - it has a local maximum at point P with f'(x) changing from positive to negative and f''(x) being negative at point P. Therefore, Option D is the best choice.

The Graphs of the First and Second Derivatives of a Function