§3.5 Global optimization

Introduction

Global Optimization

Let $f(x) = 2 + \frac{3}{1+(x+1)^2}$.

Find $f'(x)$ and use it to determine all of the critical numbers of $f$.

Find all critical numbers of $g$ that lie in the interval $-2 \le x \le 3$.

Use a graphing utility to construct the graph of $g$ on the interval $-2 \le x \le 3$.

From the graph, determine the $x$-values at which the absolute minimum and absolute maximum of $g$ occur on the interval $[-2,3]$.

How do your answers change if we instead consider the interval $-2 \le x \le 2$?

What if we instead consider the interval $-2 \le x \le 1$?

$h(x) = xe^{-x}$, $[0,3]$

$p(t) = \sin(t) + \cos(t)$, $[-\frac{\pi}{2}, \frac{\pi}{2}]$

$q(x) = \frac{x^2}{x-2}$, $[3,7]$

$f(x) = 4 - e^{-(x-2)^2}$, $(-\infty, \infty)$

$h(x) = xe^{-ax}$, $[0, \frac{2}{a}]$ ($a \gt 0$)

$f(x) = b - e^{-(x-a)^2}$, $(-\infty, \infty)$, $a, b \gt 0$