Find the intervals in which the function f(x) = 20 - 9x+6x^2- x^3 is…

The correct answer is Option (1) → Strictly decreasing on $(−∞,1]∪[3,∞)$ and strictly increasing on $[1,3]$.

Given $f(x) = 20 - 9x+6x^2-x^3, D_f= R$.

Differentiating w.r.t. $x$, we get

$f'(x) = 0-9+12x-3x^2$

$=-3(x^2-4x+3)$

$=-3(x-1) (x-3)$.

Now $f'(x) > 0$ iff $-3(x-1) (x-3) > 0$

$⇒(x-1) (x-3) <0$

$⇒ 1<x<3⇒x∈ (1,3)$

⇒ f is strictly increasing in $[1, 3]$.

And $f'(x) < 0$ iff $-3(x-1) (x-3) <0$

$⇒(x-1) (x-3) > 0$

$⇒ x < 1$ or $x > 3$

$⇒x∈ (-∞, 1) ∪ (3,∞)$

⇒ f is strictly decreasing in $(-∞, 1] ∪ [3,∞)$.