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# Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is - Mathematics

#### Question

Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is

(a) strictly increasing

(b) strictly decreasing

#### Solution

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).

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