WBJEE · Maths · Application of Derivatives
Let \(f:[a, b] \rightarrow \mathbb{R}\) be continuous in \([a, b]\), differentiable in \((a, b)\) and \(f(a)=0=f(b)\). Then
- A there exists at least one point \(c \in(a, b)\) for which \(f^{\prime}(c)=f(c)\)
- B \(f^{\prime}(x)=f(x)\) does not hold at any point of \((a, b)\)
- C at every point of \((a, b), f^{\prime}(x)>f(x)\)
- D at every point of \((a, b), f^{\prime}(x) < f(x)\)
Answer & Solution
Correct Answer
(A) there exists at least one point \(c \in(a, b)\) for which \(f^{\prime}(c)=f(c)\)
Step-by-step Solution
Detailed explanation
\(:\) Let \(g(x)=e^{-x} f(x)\) \[ \mathrm{g}(\mathrm{a})=\mathrm{g}(\mathrm{b})=0 \] By Rolle's theorem, for atleast one \(c \in(a, b)\) where \(\mathrm{g}^{\prime}(\mathrm{c})=0\) or \(e^{-c} f^{\prime}(c)-f(c) e^{-c}=0\) or \(e^{-c}\left(f^{\prime}(c)-f(c)\right)=0\) or…
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