TS EAMCET · Maths · Application of Derivatives
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?
- A Both \((A)\) and (R) are true and (R) is the correct reason for (A)
- B Both \((A)\) and \((R)\) are true, but (R) is not the correct reason for (A)
- C (A) is true, (R) is false
- D (A) is false, (R) is true
Answer & Solution
Correct Answer
(A) Both \((A)\) and (R) are true and (R) is the correct reason for (A)
Step-by-step Solution
Detailed explanation
Given, \(f(x)=x e^{-x}\) \(\begin{aligned} f^{\prime}(x) & =e^{-x}-x e^{-x} \\ f^{\prime \prime}(x) & =-e^{-x}-e^{-x}+x e^{-x} \\ & =-2 e^{-x}+x e^{-x}\end{aligned}\) For maximum, put \(f^{\prime}(1)=0 \Rightarrow x=1\) and \(f^{\prime \prime}(1)=-1 < 0\) \(\therefore\) Both…
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