Global optimization

Find the exact absolute maximum and minimum of each function on the stated interval.

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

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

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

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

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

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

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.

1. $f$ is a function such that $f''(x) \lt 0$ for every $x$…

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.

1. $p(x) = x^3 - a^2x$, $[0,a]$ ($a \gt 0$)

2. $r(x) = axe^{-bx}$, $[\frac{1}{2b}, \frac{2}{b}]$ ($a \gt 0, b \gt 1$)

3. $w(x) = a(1-e^{-bx})$, $[b, 3b]$ ($a, b \gt 0$)

4. $s(x) = \sin(kx)$, $\left[\frac{\pi}{3k}, \frac{5\pi}{6k}\right]$ ($k \gt 0$)

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 expla…

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 $s$.

1. $[\frac{\pi}{6}, \frac{7\pi}{6}]$

2. $[0, \frac{\pi}{2}]$

3. $[0, 2\pi]$

4. $[\frac{\pi}{3}, \frac{5\pi}{6}]$