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  4. The length of the longest interval in which the function f(x)=x3-3a2x+4 is decreasing is (∀ a > 0)

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Q. The length of the longest interval in which the function $f\left(x\right)=x^{3}-3a^{2}x+4$ is decreasing is $\left(\forall a > 0\right)$

NTA AbhyasNTA Abhyas 2022

Solution:

$f^{'} \left(x\right) = 3 x^{2} - 3 a^{2}$
$=3\left(x - a\right)\left(x + a\right)$
Solution
$\therefore f\left(x\right)$ decreases in $x\in \left[- a , a\right]$
i.e. length $=2a$