Answer 1

We have:

`f(x) = 3x^4 − 4x^3 −12x^2 + 5`

`Now, f'(x) = 12x^3 − 12x^2 − 24x`

`Now, f'(x) = 0`

`⇒12x^3 −12x^2−24x = 0`

`⇒12x(x^2−x−2) = 0`

`⇒12x(x^2−2x+x−2)=0`

`⇒12x[x(x−2)+1(x−2)] = 0`

`⇒12x (x+1)(x−2)=0`

`⇒x=0 ; x = −1; x = 2`

So, the points x = −1, x = 0 and x = 2 divide the real line into four disjoint intervals, namely (−∞,−1), (−1,0), (0,2) and (2,∞).

INTERVAL	SIGN OF f ' (x)=12x (x+1)(x −2)	NATURE OF FUNCTION
(−∞,−1)	(−)(−)(−)=−or<0	Strictly decreasing
(−1,0)	(−)(+)(−)=+or>0	Strictly increasing
(0,2)	(+)(+)(−) = − or<0	Strictly decreasing
(2,∞)	(+)(+)(+) = + or >0	Strictly increasing

(a) The given function is strictly increasing in the intervals (−1,0) ∪ (2,∞).
(b) The given function is strictly decreasing in the intervals (−∞,−1) ∪ (0,2).