§3.5 Global optimization

Introduction

Global Optimization

Let $f(x) = 2 + \frac{3}{1+(x+1)^2}$. Writing the fraction as $3(1+(x+1)^2)^{-1}$, differentiation gives $f'(x) = \frac{-6(x+1)}{(1+(x+1)^2)^2}$. The critical numbers and global behavior of $f$ are

Suppose $g'(x) = x^2 - 2$. On the interval $[-2, 3]$, the critical numbers $\pm\sqrt{2}$ split the interval, and comparing $g(-2)$, $g(-\sqrt{2})$, $g(\sqrt{2})$, $g(3)$ shows the absolute maximum occurs at

For the same $g$ restricted instead to $[-2, 2]$, the absolute extrema occur at

For the same $g$ on the interval $[-2, 1]$, the absolute extrema occur at

For $h(x) = xe^{-x}$ on $[0, 3]$, the only critical number is $x = 1$. Comparing $h(0)$, $h(1)$, and $h(3)$, the absolute maximum value is

For $p(t) = \sin(t) + \cos(t)$ on $[-\frac{\pi}{2}, \frac{\pi}{2}]$, the critical number is $t = \frac{\pi}{4}$. Comparing the three candidate values shows

For $q(x) = \frac{x^2}{x-2}$ on $[3, 7]$, the only critical number in the interval is $x = 4$. Comparing $q(3)$, $q(4)$, and $q(7)$ shows

For $f(x) = 4 - e^{-(x-2)^2}$ on all real numbers, which statement is correct?

For $h(x) = xe^{-ax}$ on $[0, \frac{2}{a}]$ with $a \gt 0$, the only critical number is $x = \frac{1}{a}$. The absolute extrema are

For $f(x) = b - e^{-(x-a)^2}$ with $a, b \gt 0$ on all real numbers, the function has

For $q(x) = \frac{x^2}{x-2}$ on $[3, 7]$, compute the absolute maximum value as a decimal.

For $p(t) = \sin(t) + \cos(t)$ on $[-\frac{\pi}{2}, \frac{\pi}{2}]$, compute the absolute maximum value, rounded to three decimals.

A continuous function on a closed interval $[a, b]$ attains its absolute extrema either at the endpoints or at critical numbers inside the interval.