Observe the statements given below :Assertion | vedclass

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(A) Given function is $f(x) = x e^{-x}$.First,we find the first derivative: $f^{\prime}(x) = (1)e^{-x} + x(-e^{-x}) = e^{-x}(1-x)$.Setting $f^{\prime}

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Both $(A)$ and $(R)$ are true and $(R)$ is the correct reason for $(A)$

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(A) Given function is $f(x) = x e^{-x}$.First,we find the first derivative: $f^{\prime}(x) = (1)e^{-x} + x(-e^{-x}) = e^{-x}(1-x)$.Setting $f^{\prime}(x) = 0$,we get $1-x = 0$,which implies $x = 1$.Next,we find the second derivative: $f^{\prime \prime}(x) = -e^{-x}(1-x) + e^{-x}(-1) = e^{-x}(x-1-1) = e^{-x}(x-2)$.Evaluating at $x=1$: $f^{\prime \prime}(1) = e^{-1}(1-2) = -e^{-1} = -\frac{1}{e}$.Since $f^{\prime \prime}(1) Thus,both Assertion $(A)$ and Reason $(R)$ are true,and $(R)$ is the correct explanation for $(A)$.

Observe the statements given below :
Assertion $(A)$ : $f(x)=x e^{-x}$ has the maximum at $x=1$
Reason $(R)$ : $f^{\prime}(1)=0$ and $f^{\prime \prime}(1) < 0$
Which of the following is correct?

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Observe the statements given below :Assertion $(A)$ : $f(x)=x e^{-x}$ has the maximum at $x=1$Reason $(R)$ : $f^{\prime}(1)=0$ and $f^{\prime \prime}(1) < 0$Which of the following is correct?