Question

Let \(f: \mathbb{R} \to \mathbb{R}\) be continuous on \(\mathbb{R}\) and differentiable on \((- \infty, 0) \cup (0, \infty)\). Which of the following statements is (are) always TRUE?

• A. If \(f\) is differentiable at 0 and \(f'(0) = 0\), then \(f\) has a local maximum or a local minimum at 0.
• B. If \(f\) has a local minimum at 0, then \(f\) is differentiable at 0 and \(f'(0) = 0\).
• C. If \(f'(x)<0\) for all \(x<0\) and \(f'(x)>0\) for all \(x>0\), then \(f\) has a global maximum at 0.
• D. If \(f'(x)>0\) for all \(x<0\) and \(f'(x)<0\) for all \(x>0\), then \(f\) has a global maximum at 0.

Hint:

To confirm the location of local or global extrema, check the sign change in the derivative at the point of interest. If the derivative changes from positive to negative, a local maximum is likely.

Answer 1

The Correct Option is D

Step 1: Analyzing the properties of the function.
For \(f\) to have a global maximum at 0, the derivative of the function must change sign from positive to negative at 0, which is described in option (D). This indicates a peak at 0, where the function reaches its highest point.
Step 2: Analyzing the options.
(A) If \(f\) is differentiable at 0 and \(f'(0) = 0\), then \(f\) has a local maximum or a local minimum at 0: This is not necessarily true. A derivative of zero at a point is a necessary condition for a local extremum, but it does not guarantee that the point is a maximum or minimum.
(B) If \(f\) has a local minimum at 0, then \(f\) is differentiable at 0 and \(f'(0) = 0\): This is true in some cases, but not always. The condition \(f'(0) = 0\) holds at local minima, but differentiability is not required at the minimum itself in every case.
(C) If \(f'(x)<0\) for all \(x<0\) and \(f'(x)>0\) for all \(x>0\), then \(f\) has a global maximum at 0: This is incorrect. While the derivative conditions indicate a local minimum, they do not guarantee a global maximum.
(D) If \(f'(x)>0\) for all \(x<0\) and \(f'(x)<0\) for all \(x>0\), then \(f\) has a global maximum at 0: This is correct. The change in sign of the derivative from positive to negative implies a global maximum at 0.
Step 3: Conclusion.
The correct answer is \((D)\), as it describes a function with a global maximum at 0 based on the behavior of its derivative.