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प्रश्न
Find the intervals in which the function f(x) = 3x4 − 4x3 − 12x2 + 5 is
(a) strictly increasing
(b) strictly decreasing
उत्तर
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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