Find the intervals in which the function $f$ given by $f(x)=4x^3−6x^2-72x+30$ is (i) strictly increasing (ii) strictly decreasing.

The correct answer is During $(−∞, -2)$ and $(3, ∞)$, the function is strictly increasing.
During $(-2, 3)$, the function is strictly decreasing.

$f(x)=4x^3−6x^2-72x+30$


Differentiate $f'(x)=12x^2−12x-72$


$f'(x)=0$ ⇒ $0=12x^2−12x-72$

⇒ $12x^2−12x-72=0$

⇒ $x^2−x-6=0$

⇒ $x^2−3x+2x-6=0$

⇒ $x(x−3)+2(x-3)=0$

⇒ $(x−3)(x+2)=0$

∴ $x = 3, -2$



During the interval $(−∞, -2)$ and $(3, ∞)$, $12x^2−12x-72 > 0$ the function is strictly increasing.

During the interval $(-2, 3)$, $12x^2−12x-72 < 0$, the function is strictly decreasing.