.

2.4 Mean value theorem

1.

(a) $${\xi = \frac{3}{2}}$$ (b) $${\xi = \sqrt{\frac{13}{3}}}$$ (c) $${\xi = \frac{1}{\sqrt{2}}}$$

2.

(a) Concave up on $(-\infty, 0)$, concave down on $(0, \infty)$, local minimum $4$ at $x = -1$, local minimum 0 at $x = 1$, inflection point $(0,2)$

(b) Concave up on $(-\infty, 0)$, concave down on $(0, \infty)$, locla maximum $-2$ at $x = -1$, local minimum $2$ at $x = 1$

(c) Concave up on $(-\infty, -1)$, concave down on $(-1, \infty)$, local maximum $\frac{2\sqrt{3}}{9}$ at $x = -1 -\frac{\sqrt{3}}{3}$ local minimum $-\frac{2\sqrt{3}}{9}$ at $x = -1 + \frac{\sqrt{3}}{3}$, inflection point $(-1,0)$

(d) Concave up on $(-\infty, -\sqrt{3}), (0, \sqrt{3})$, concave down on $(-\sqrt{3}, 0), (\sqrt{3},\infty)$, local minimum $-\frac{1}{2}$ at $x = -1$, local maximum $\frac{1}{2}$ at $x = 1$, $(-\sqrt{3}, -\frac{\sqrt{3}}{4}), (0,0), (\sqrt{3}, \frac{\sqrt{3}}{4})$ are points of inflection.

(e) concave up on $(-2,1)$, concave down on $(-\infty,-2), (1,\infty)$, local minimum 0 at $x = -2$ and $x = 1$, local maximum $\frac{9}{4}$ at $x = -\frac{1}{2}$.

3.

(a) $400$ (b) $${\frac{32\sqrt{3}}{9}}$$ (c) $${\frac{64\sqrt{2}}{3}}$$ (d) $${\frac{1}{2}}$$