Question

Let \(\mathrm{f}:[\mathrm{a}, \mathrm{b}] \rightarrow \mathbb{R}\) be such \(\mathrm{f}\) is differentiable in \((\mathrm{a}, \mathrm{b}), \mathrm{f}\) is continuous at \(\mathrm{x}=\mathrm{a} \& \mathrm{x}=\mathrm{b}\) and moreover \(\mathrm{f}(\mathrm{a})=0=\mathrm{f}(\mathrm{b}) .\) Then
(A) there exists at least one point \(c\) in \((a, b)\) such that \(f^{\prime}(c)=f(c)\)
(B) \(f^{\prime}(x)=f(x)\) does not hold at any poit in \((a, b)\)
(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 1

Ans : (A)
Hint : Let, \(g(x)=e^{-x} f(x)\) which is continuous in [a, b] and differentiable in (a, b)
Now, \(g(a)=g(b)=0\)
\(\Rightarrow \exists k \in(a, b)\) so that
\(g^{\prime}(k)=0 \Rightarrow e^{-k} f^{\prime}(k)-e^{-k} f(k)=0\)
\(\Rightarrow f^{\prime}(k)=f(k)\)

Answer 2

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