Consider the function $f(x)=\begin{cases}-x^3+3x^2+1, & x\le2 \\ \cos x, & 24\end{cases}$ Which of the following statements is true?
- A. f(x) has a local maximum at x=1 which is also global
- B. f(x) has a local maximum at x=2 which is not global
- C. f(x) has a local maximum at x=pi which is not global
- D. f(x) has a global maximum at x=0
✅ Correct Answer: f(x) has a local maximum at x=2 which is not global
Explanation
For $x \le 2$ $f'(x) = -3x^2 + 6x$ $f'(x)=0$ gives $x=0,2$ $f''(x) = -6x + 6$ $f''(2) < 0$ so $x=2$ is a local maximum Comparing values across intervals shows it is not the global maximum
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