TS EAMCET · Maths · Differential Equations
Let \(f:[2,5] \rightarrow \mathbf{R}\) be a differentiatiable function and \(\frac{f(5)}{f(2)}=1\). If there is a \(c \in(2,5)\) such that \(c f^{\prime}(c)=2 f(c)-2 c^3\), then \(f(x)=\)
- A \(-2 x^3+\frac{78}{7} x^2\)
- B \(x^3-8 x^2+17 x-10\)
- C \(x^3-6 x^2+3 x+10\)
- D \(x^3-7 x^2+10 x\)
Answer & Solution
Correct Answer
(A) \(-2 x^3+\frac{78}{7} x^2\)
Step-by-step Solution
Detailed explanation
Given, \(c f^{\prime}(c)=2 f(c)-2 c^3=x f^{\prime}(x)=2 f(x)-2 x^3\) Put, \(\quad f(x)=y \Rightarrow f^{\prime}(x)=\frac{d y}{d x}\) \(\because \quad x \frac{d y}{d x}-2 y=-2 x^3\) \(\frac{d y}{d x}-\frac{2 y}{x}=-2 x^2\) Here IF…
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