91Ó°ÊÓ

A function \(f\) and its domain are given. Determine the critical points, evaluate \(f\) at these points, and find the (global) maximum and minimum values. \(f(z)=\frac{1}{z^{2}} ;\left[-2,-\frac{1}{2}\right]\)

Use the Bisection Method to approximate the real root of the given equation on the given interval. Each answer should be accurate to two decimal places. $$ 2 \cos x-\sin x=0 ;[1,2] $$

A function is defined and a closed in terval is given. Decide whether the Mean Value Theorem applies to the given function on the given interval. If it does, find all possible values of \(c ;\) if not, state the reason. In each problem, sketch the graph of the given function on the given interval. $$ f(x)=x^{2}+x ;[-2,2] $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=3 x^{2}+\sqrt{3} $$

Find two numbers whose product is -12 and the sum of whose squares is a minimum.

Use the Bisection Method to approximate the real root of the given equation on the given interval. Each answer should be accurate to two decimal places. $$ x-2+2 \cos x=0 ;[1,2] $$

Show that the indicated function is a solution of the given differential equation; that is, substitute the indicated function for y to see that it produces an equality. $$\left(\frac{d y}{d x}\right)^{2}+y^{2}=1 ; y=\sin (x+C)\( and \)y=\pm 1$$

A function is defined and a closed in terval is given. Decide whether the Mean Value Theorem applies to the given function on the given interval. If it does, find all possible values of \(c ;\) if not, state the reason. In each problem, sketch the graph of the given function on the given interval. $$ g(x)=(x+1)^{3} ;[-1,1] $$

Identify the critical points. Then use (a) the First Derivative Test and (if possible) (b) the Second Derivative Test to decide which of the critical points give a local maximum and which give a local minimum. $$ f(x)=\frac{1}{2} x+\sin x, 0

Identify the critical points and find the maximum value and minimum value on the given interval. $$ f(x)=x^{2}+4 x+4 ; I=[-4,0] $$