AP EAMCET · Maths · Differentiation
If \(y=\frac{\log x}{x}\), then the value of \(x^2 \frac{d^2 y}{d x^2}+3 x \frac{d y}{d x}+y\) at the point \((\sqrt[3]{e}, \sqrt{e})\) is
- A 0
- B 1
- C \(\mathrm{e}\)
- D \(2 \mathrm{e}\)
Answer & Solution
Correct Answer
(A) 0
Step-by-step Solution
Detailed explanation
\begin{aligned} & \text { } y=\frac{\log x}{x} \\ & \Rightarrow \frac{d y}{d x}=\frac{1-\log x}{x^2} \\ & \text { and } \frac{d^2 y}{d x^2}=\frac{-3 x+2 x \log x}{x^4} \\ & \text { Now Consider, } \\ & x^2 \frac{d^2 y}{d x^2}+3 x \frac{d y}{d x}+y \\ & =x^2\left(\frac{-3 x+2 x…
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