Question 46

Notice that $f(x) = -\left(x - \frac{|x|}{2}\right)^2$. Splitting into $x \ge 0$ ($|x| = x$) and $x < 0$ ($|x| = -x$):

1. Extrema: Since $f(0) = 0$ and $f(x) < 0$ for all $x \neq 0$, $f$ has a strict local (and global) maximum at $x = 0$ (Option (A) is true) and is strictly increasing for $x < 0$ and strictly decreasing for $x > 0$, so it has no local minimum (Option (B) is false).
2. First derivative $f'(x)$:
   $$f'(x) = \begin{cases} -\frac{x}{2}, & x \ge 0 \\ -\frac{9x}{2}, & x < 0 \end{cases}$$
   Since $\lim_{x \to 0^+} f'(x) = 0 = \lim_{x \to 0^-} f'(x) = f'(0)$, $f'$ is continuous over all of $\mathbb{R}$ (Option (C) is true).
3. Differentiability of $f'$: At $x = 0$, the right-hand derivative of $f'$ is $-1/2$ while the left-hand derivative of $f'$ is $-9/2$. Since $-1/2 \neq -9/2$, $f'$ is not differentiable at $x = 0$ (Option (D) is true).