19) The interval in which y = x ^2 e ^{-x}is increasing is

(A) ( - \infty , \infty ) (B) ( - 2 , 0 )(C) ( - 2 , \infty ) (D) ( 0, 2 )

Answers (1)

Given function is,
f(x) \Rightarrow y = x ^2 e ^{-x}
f^{'}(x) \Rightarrow \frac{dy}{dx} = 2x e ^{-x} + -e^{-x}(x^{2})
                            xe ^{-x}(2 -x)
f^{'}(x) = xe ^{-x}(2 -x)
Now, it is clear that f^{'}(x) > 0  only in the interval (0,2)
So,  f(x) \Rightarrow y = x ^2 e ^{-x} is  an increasing function for the interval  (0,2)
Hence, (D) is the answer

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