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 $x = 1$

=2.6zw =1(c) 上に凸 $(-\infty, -1)$，下に凸 $(-1, \infty)$, $${x = -1 -\frac{\sqrt{3}}{3}}$$で極大値 $${\frac{2\sqrt{3}}{9}}$$， $${x = -1 + \frac{\sqrt{3}}{3}}$$で極小値 $${-\frac{2\sqrt{3}}{9}}$$, 変曲点$(-1,0)$

=2.6zw =1(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

=2.6zw =1(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) $${\frac{32\sqrt{3}}{9}}$$ =2.6zw =1(c) $${\frac{64\sqrt{2}}{3}}$$ (d) $${\frac{1}{2}}$$