Chapter 4 - Applications of the Derivative - 4.2 What Derivatives Tell Us - 4.2 Exercises - Page 257: 33

$f$ is increasing on $(1, 4)$ and is decreasing on $(−∞, 1)$ and on $(4,∞)$.

$f'(x) = −60x^4+300x^3−240x^2$ $ = −60x^2(x^2−5x+4) $ = $−60x^2(x−4)(x−1)$. This is $0$ for $x = 0$, $x = 1$, and $x = 4$. Note that $f'(−1) = −600 < 0$, $f'(1/2) ≈ −26.25 < 0$, $f'(2) = 480 > 0$, and $f'(5) = −6000 < 0$. Thus $f$ is increasing on $(1, 4)$ and is decreasing on $(−∞, 1)$ and on $(4,∞)$.