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

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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).

  Is there an error in this question or solution?
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Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is Concept: Increasing and Decreasing Functions.
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