Question
Find the intervals in which the function f given by f(x) = 2x3 – 3x2 – 36x + 7 is increasing.
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Answer
It is given that function f(x) = 2x3 - 3x2 - 36x + 7
$\Rightarrow$ f'(x) = 6x2 - 6x - 36
$\Rightarrow$ f'(x) = 6(x2 - x - 6)
$\Rightarrow$ f'(x) = 6(x + 2) (x - 3)
$\Rightarrow$ f'(x) = 6(x + 2)(x - 3)
If f'(x) = 0, then we get,
$\Rightarrow$ x = -2, 3
So, the points x = -2 and x = 3 divides the real line into two disjoint intervals, $(-\infty, -2),(-2,3) $ and $(3, \infty)$
So, in interval $(-\infty, -2) and (3, \infty)$
$f^\prime$(x) = 6(x + 2)(x - 3) > 0
But, in $(-2, 3)$, $f^\prime(x)<0$
Therefore, the given function 'f' is strictly increasing in interval $(-\infty, 2)and(3, \infty)$ while as strictly decreasing in $(-2, 3)$
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