Exercise 8

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