Question

Consider the function \( f(x) = \begin{cases} -x^3 + 3x^2 + 1, & \text{if } x \leq 2 \\ \cos(x), & \text{if } 2 < x \leq 4 \\ e^{-x}, & \text{if } x > 4 \end{cases} \) Which of the following statements about \( f(x) \) is true:

• A. \( f(x) \) has a local maximum at \( x = 1 \), which is also the global maximum.
• B. \( f(x) \) has a local maximum at \( x = 2 \), which is not the global maximum.
• C. \( f(x) \) has a local maximum at \( x = \pi \), but it is not the global maximum.
• D. \( f(x) \) has a global maximum at \( x = 0 \).

Hint:

To find local maxima or minima, take the derivative of the function, set it equal to zero, and analyze the second derivative.

Answer 1

The Correct Option is B

Step 1: Examine the first part of the function. For \( x \leq 2 \), the function is: \[ f(x) = -x^3 + 3x^2 + 1. \] Taking the derivative: \[ f'(x) = -3x^2 + 6x. \] Setting \( f'(x) = 0 \) to find critical points: \[ -3x^2 + 6x = 0 \quad \Rightarrow \quad x(x - 2) = 0. \] Thus, the critical points are \( x = 0 \) and \( x = 2 \). For \( x = 0 \), \( f''(x) = 6 \), indicating a local minimum.
For \( x = 2 \), \( f''(x) = -6 \), indicating a local maximum.
Step 2: Examine the second part of the function. For \( 2 < x \leq 4 \), the function is: \[ f(x) = \cos(x). \] The derivative is: \[ f'(x) = -\sin(x). \] Setting \( f'(x) = 0 \) gives \( \sin(x) = 0 \), which occurs at \( x = \pi \). This is a local maximum since \( \cos(x) \) has a maximum at \( x = \pi \). 
Step 3: Examine the third part of the function. For \( x > 4 \), the function is: \[ f(x) = e^{-x}. \] This function is always decreasing, so no maximum exists here. 
Step 4: Conclusion. The function has a local maximum at \( x = 2 \) for the first piece, but it is not the global maximum because the global maximum occurs at \( x = \pi \) for the second piece of the function, where \( \cos(x) \) attains its maximum value. Thus, the correct answer is: \[ \boxed{(2) \, f(x) \text{ has a local maximum at } x = 2, \text{ which is not the global maximum.}}. \]