The length of the longest interval in which the function f x = x 3 - 3 a 2 x + 4 is decreasing is ∀ a > 0

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The length of the longest interval in which the function

f (x) = x^{3} - 3 a^{2} x + 4

is decreasing is

(\forall a > 0)

MathematicsApplication of DerivativesJEE Main

Options:

A $a$
B $2 a$
C $3 a$
D $4 a$

Solution:

1683 Upvotes Verified Answer

The correct answer is:

2 a

f^{'} (x) = 3 x^{2} - 3 a^{2}

= 3 (x - a) (x + a)

∴ f (x)

decreases in

x \in [- a, a]

i.e. length

= 2 a

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