91Ó°ÊÓ

Each of the graphs of the functions has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. $$ f(x)=x^{3}+6 x^{2}+9 x $$

Each of the graphs of the functions has one relative maximum and one relative minimum point. Plot these two points and check the concavity there. Using only this information, sketch the graph. $$ f(x)=\frac{1}{9} x^{3}-x^{2} $$

Use the given information to make a good sketch of the function \(f(x)\) near \(x=3\). $$ \begin{array}{l} f(3)=3, f^{\prime}(3)=1, \text { inflection point at } x=3, f^{\prime \prime}(x)<0\\\ \text { for } x>3 \end{array} $$

Find the dimensions of the rectangular garden of greatest area that can be fenced off (all four sides) with 300 meters of fencing.

Find two positive numbers, \(x\) and \(y\), whose sum is 100 and whose product is as large as possible.

A closed rectangular box is to be constructed with a base that is twice as long as it is wide. If the total surface area must be 27 square feet, find the dimensions of the box that will maximize the volume.

Sketch the graphs of the following functions for \(x>0\). $$ y=\frac{1}{x^{2}}+\frac{x}{4}-\frac{5}{4}[\text { Hint }:(1,0) \text { is an } x \text { -intercept.] } $$

Consider a smooth curve with no undefined points. (a) If it has two relative maximum points, must it have a relative minimum point? (b) If it has two relative extreme points, must it have an inflection point?

In a medical experiment, the body weight of a baby rat in the control group after \(t\) days was \(f(t)=4.96+.48 t+.17 t^{2}-.0048 t^{3}\) grams. (Source: Growth, Development and Aging.) (a) Graph \(f(t)\) in the window \([0,20]\) by \([-12,50]\). (b) Approximately how much did the rat weigh after 7 days? (c) Approximately when did the rat's weight reach 27 grams? (d) Approximately how fast was the rat gaining weight after 4 days? (e) Approximately when was the rat gaining weight at the rate of 2 grams per day? (f) Approximately when was the rat gaining weight at the fastest rate?

If the lungs are less flexible than normal, an increase in pressure will cause a smaller change in lung volume than in a normal lung. In this case, is the compliance relatively high or low?