91Ó°ÊÓ

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. $$ g(x)=|x| ;[-2,2] $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=x-4 $$

For what number does the principal square root exceed eight times the number by the largest amount?

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. $$ f(x)=x^{3}-12 x+\pi $$

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. \(f(t)=\frac{1}{t} ;[1,4]\)

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. $$ x^{4}+5 x^{3}+1=0 ;[-1,0] $$

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. $$ f(\theta)=\sin 2 \theta, 0<\theta<\frac{\pi}{4} $$

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. \(f(z)=\frac{1}{z^{2}} ;\left[-2,-\frac{1}{2}\right]\)

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. $$ f(x)=x^{2}+x ;[-2,2] $$

In Problems \(I-10\), use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ h(t)=t^{2}+2 t-3 $$