JEE Main20264 April 2026Evening ShiftMathematicsApplication of DerivativesActual

_ 0 x (16 ( x 2 ) ^3 ( x 2 ) ) is equal to:

Options

A3 3 2
B3 3
C4 3
D6 3

Correct answer

B. 3 3

Step-by-step solution

Let y = x 2 . Since 0 x , we have 0 y 2 . The given expression becomes f(y) = 16 y ^3 y . Differentiating with respect to y : f'(y) = 16( y ^3 y + y 3 ^2 y(- y)) f'(y) = 16 ^2 y( ^2 y - 3 ^2 y) For maximum value, f'(y) = 0 : ^2 y - 3 ^2 y = 0 ^2 y = 1 3 Since y [0, 2 ] , y = 1 3 y = 6 . Substituting y = 6 in f(y) : f ( 6 ) = 16 ( 6 ) ^3 ( 6 ) f ( 6 ) = 16 1 2 ( 3 2 )^3 f ( 6 ) = 8 3 3 8 = 3 3 The maximum value is 3 3 . Answer: 3 3

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