91Ó°ÊÓ

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

The first and second derivatives of the function \(f(x)\) have the values given in Table 1. (a) Find the \(x\) -coordinates of all relative extreme points. (b) Find the \(x\) -coordinates of all inflection points. $$\text { Table 1 Values of the First Two Derivatives of a Function }$$ $$\begin{array}{ccc} \hline x & f^{\prime}(x) & f^{\prime \prime}(x) \\ \hline 0 \leq x < 2 & \text { Positive } & \text { Negative } \\ 2 & 0 & \text { Negative } \\ 2 < x < 3 & \text { Negative } & \text { Negative } \\ 3 & \text { Negative } & 0 \\ 3 < x < 4 & \text { Negative } & \text { Positive } \\ 4 & 0 & 0 \\ 4 < x \leq 6 & \text { Negative } & \text { Negative } \\ \hline \end{array}$$

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

Consider a parabolic arch whose shape may be represented by the graph of \(y=9-x^{2},\) where the base of the arch lies on the \(x\) -axis from \(x=-3\) to \(x=3 .\) Find the dimensions of the rectangular window of maximum area that can be constructed inside the arch.

Sketch the graph of a function having the given properties. Defined and increasing for all \(x \geq 0 ;\) inflection point at \(x=5\) asymptotic to the line \(y=\left(\frac{2}{4}\right) x+5\)