Let f(x) = x 3 e –3x , x ^gt 0. Then the maximum value of f(x) is

Explanation: f ( x ) = x 3 ⋅ e − 3 x = f ′ ( x ) = 3 x 2 e − 3 x + x 3 e − 3 x ( − 3 ) = x 2 3 e − 3 x [ 1 − x ] = 0 , x = 1 Maximum at x = 1 f ( 1 ) = e − 3

Question:

Let f(x) = x³e^–3x, x ^gt 0. Then the maximum value of f(x) is

WBJEE

Updated On: Aug 9, 2024

(A) e^–3
(B) 3e^–3
(C) 27e^–9
(D) ∞

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The Correct Option is A

Solution and Explanation

Explanation:

f (x) = x^{3} \cdot e^{- 3 x}

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

= x^{2} 3 e^{- 3 x} [1 - x] = 0, x = 1

Maximum at

x = 1

f (1) = e^{- 3}

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