2.4 Mean value theorem

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

(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 $(-1, \infty)$ , $\displaystyle{x = -1 -\frac{\sqrt{3}}{3}}$ で極大値 $\displaystyle{\frac{2\sqrt{3}}{9}}$ ， $\displaystyle{x = -1 + \frac{\sqrt{3}}{3}}$ で極小値 $\displaystyle{-\frac{2\sqrt{3}}{9}}$ , 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 $\displaystyle{-\frac{1}{2}}$ at $x = -1$ ，local maximum $\displaystyle{\frac{1}{2}}$ at $x = 1$ , $\displaystyle{(-\sqrt{3}, -\frac{\sqrt{3}}{4}), (0,0), (\sqrt{3}, \frac{\sqrt{3}}{4})}$ 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 $x = -\frac{1}{2}$

(a) $400$ (b) $\displaystyle{\frac{32\sqrt{3}}{9}}$ =2.6zw =1(c) $\displaystyle{\frac{64\sqrt{2}}{3}}$ (d) $\displaystyle{\frac{1}{2}}$