#### Question

Find the intervals in which the function f(*x*) = 3*x*^{4} − 4*x*^{3} − 12*x*^{2} + 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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