If $f(x)=sin(\frac{x}{2}),$ then there exists a number $c$ in the interval $e\frac{2}{3}\frac{3}{4}\frac{5}{6}{x}^2+2y=0(0,-\frac{1}{2})y\frac{1}{2}-\frac{1}{2}xt0x(t)=t^3-3t^2-9t+1t\underset{x0}{lim}\frac{1-co{s}^2(2x)}{x^2}xf(x)=(x-2)(x-3{)}^2-\frac{7}{3}-\frac{5}{2}\frac{7}{3}-\frac{7}{8}-\frac{7}{24}\frac{7}{24}\frac{7}{8}\frac{}{2}$ that satisfies the conclusion $of$ the Mean Value Theorem. Which $of$ the following could $bee?$

$(a)\frac{2}{3}$

$(b)\frac{3}{4}$

$(c)\frac{5}{6}$

$(d)$

The point $on$ the curve $x^2+2y=0$ that $is$ nearest the point $(0,-\frac{1}{2})$ occurs where $yis$

$(a)\frac{1}{2}$

$(b)0$

$(c)-\frac{1}{2}$

$(d)-1$

A particle moves aloug the $x-$axis $so$ that $at$ any time $t0$ its position $is$ given $byx(t)=t^3-3t^2-9t+1.$ For what values $oftis$ the particle $at$ rest?

$(a)No$ values

$(b)1$ ouly

$(c)3$ only

$(d)5$ only

$\underset{x0}{lim}\frac{1-co{s}^2(2x)}{x^2}is$

$(a)1$

$(b)0$

$(c)2$

$(d)4$

For what value $ofx$ does the function $f(x)=(x-2)(x-3{)}^2$ have a relative maximum?

$(a)-3$

$(b)-\frac{7}{3}$

$(c)-\frac{5}{2}$

$(d)\frac{7}{3}$

The top $ofa25-$foot ladder $is$ sliding down a vertical wall $at$ a constant rate $of3$ feet per minute. When the top $of$ the ladder $is7$ feet from the ground, what $is$ the rate $of$ change $of$ the distance between the bottom $of$ the ladder and the wall?

$(a)-\frac{7}{8}$ feet per minute

$(b)-\frac{7}{24}$ feet per minute

$(c)\frac{7}{24}$ feet per minute

$(d)\frac{7}{8}$ feet per minute All please ;(