§3.5 Global optimization (continued)

Moving toward applications

Summary

A 20 cm piece of wire is cut into two pieces. One piece is used to form a square and the other to form an equilateral triangle. How should the wire be cut to maximize the total area enclosed by the square and triangle? to minimize the area?

Draw a labeled diagram that shows the given information. What variable should we introduce to represent the choice we make in creating the box? Label the diagram appropriately with the variable, and write a sentence to state what the variable represents.

Determine a formula for the function $V$ (that depends on the variable in (a)) that tells us the volume of the box.

What is the domain of the function $V$? That is, what values of $x$ make sense for input? Are there additional restrictions provided in the problem?

Determine all critical numbers of the function $V$.

Evaluate $V$ at each of the endpoints of the domain and at any critical numbers that lie in the domain.

What is the maximum possible volume of the box? the minimum?

Based on the given information about each function, decide whether the function has global maximum, a global minimum, neither, both, or that it is not possible to say without more information. Assume that each function is twice differentiable and defined for all real numbers, unless noted otherwise. In each case, write one sentence to explain your conclusion. -$f$ is a function such that $f''(x) \lt 0$ for every $x$. -$g$ is a function with two critical numbers $a$ and $b$ (where $a \lt b$), and $g'(x) \lt 0$ for $x \lt a$, $g'(x) \lt 0$ for $a \lt x \lt b$, and $g'(x) \gt 0$ for $x \gt b$. -$h$ is a function with two critical numbers $a$ and $b$ (where $a \lt b$), and $h'(x) \lt 0$ for $x \lt a$, $h'(x) \gt 0$ for $a \lt x \lt b$, and $h'(x) \lt 0$ for $x \gt b$. In addition, $\lim_{x \to \infty} h(x) = 0$ and $\lim_{x \to -\infty} h(x) = 0$. -$p$ is a function differentiable everywhere except at $x = a$ and $p''(x) \gt 0$ for $x \lt a$ and $p''(x) \lt 0$ for $x \gt a$.

For each family of functions that depends on one or more parameters, determine the function's absolute maximum and absolute minimum on the given interval. -$p(x) = x^3 - a^2x$, $[0,a]$ ($a \gt 0$) -$r(x) = axe^{-bx}$, $[\frac{1}{2b}, \frac{2}{b}]$ ($a \gt 0, b \gt 1$) -$w(x) = a(1-e^{-bx})$, $[b, 3b]$ ($a, b \gt 0$) -$s(x) = \sin(kx)$, $\left[\frac{\pi}{3k}, \frac{5\pi}{6k}\right]$ ($k \gt 0$)

For each of the functions described below (each continuous on $[a,b]$), state the location of the function's absolute maximum and absolute minimum on the interval $[a,b]$, or say there is not enough information provided to make a conclusion. Assume that any critical numbers mentioned in the problem statement represent all of the critical numbers the function has in $[a,b]$. In each case, write one sentence to explain your answer. -$f'(x) \le 0$ for all $x$ in $[a,b]$-$g$ has a critical number at $c$ such that $a \lt c\lt b$ and $g'(x) \gt 0$ for $x \lt c$ and $g'(x) \lt 0$ for $x \gt c$-$h(a) = h(b)$ and $h''(x) \lt 0$ for all $x$ in $[a,b]$-$p(a) \gt 0$, $p(b) \lt 0$, and for the critical number $c$ such that $a \lt c \lt b$, $p'(x) \lt 0$ for $x \lt c$ and $p'(x) \gt 0$ for $x \gt c$

Let $s(t) = 3\sin(2(t-\frac{\pi}{6})) + 5$. Find the exact absolute maximum and minimum of $s$ on the provided intervals by testing the endpoints and finding and evaluating all relevant critical numbers of $s$. -$[\frac{\pi}{6}, \frac{7\pi}{6}]$-$[0, \frac{\pi}{2}]$-$[0, 2\pi]$-$[\frac{\pi}{3}, \frac{5\pi}{6}]$