91Ó°ÊÓ

In each of Exercises \(1-6,\) a function \(f\) is given. Locate each point \(c\) for which \(f(c)\) is a local extremum for \(f\). (Calculus is not needed for these exercises.) $$ f(x)=(x-2)^{2}+3 $$

Solve each of the maximum-minimum problems. Some may not have a solution, whereas others may have their solution at the endpoint of the interval of definition. What positive number plus its reciprocal gives the least sum?

In each of Exercises \(1-6,\) a function \(f\) is given. Locate each point \(c\) for which \(f(c)\) is a local extremum for \(f\). (Calculus is not needed for these exercises.) $$ f(x)=1 /\left(x^{2}+1\right) $$

Solve each of the maximum-minimum problems. Some may not have a solution, whereas others may have their solution at the endpoint of the interval of definition. Find the longest vertical chord between the curves \(y=\) \(x^{3} / 3-3 x^{2} / 2+2 x-3 / 4\) and \(y=3 x^{3} / 4-15 x^{2}+5 x, 0 \leq x \leq 3\).

Use the first derivative to determine the intervals on which the given function \(f\) is increasing and on which \(f\) is decreasing. At each point \(c\) with \(f^{\prime}(c)=0,\) use the First Derivative Test to determine whether \(f(c)\) is a local maximum value, a local minimum value, or neither. $$ f(x)=4 x^{3}-6 x^{2}+8 $$

Follow the outline given in this section to give a careful sketch of the graph of each of the functions in Exercises \(1-24\). Your sketch should exhibit, and have labeled, all of the following: a) local and global extrema, b) inflection points, c) intervals on which function is increasing or decreasing, d) intervals on which function is concave up or concave down, e) horizontal and vertical asymptotes. $$ f(x)=x^{5}-6 x^{4}+8 x^{3} $$

Calculate the indefinite integral. $$ \int(x+1)^{2} d x $$

Calculate the indefinite integral. $$ \int \csc ^{2}(x) d x $$

Determine the intervals on which the given function \(f\) is concave up, the intervals on which \(f\) is concave down, and the points of inflection of \(f\). Find all critical points. Use the Second Derivative Test to identify the points \(x\) at which \(f(x)\) is a local minimum value and the points at which \(f(x)\) is a local maximum value. $$ f(x)=(x+1)^{2} / x $$

Sketch the graph of each of the functions in Exercise \(25-40,\) exhibiting and labeling: a) all local and globa extrema; b) inflection points; c) intervals on which the func tion is increasing or decreasing; d) intervals on which the function is concave up or concave down; e) all horizontal an vertical asymptotes. $$ f(x)=x^{1 / 3} /(x-4) $$