WBJEE · Maths · Continuity and Differentiability
Let \(f:[a, b] \rightarrow R\) be such that \(f\) is differentiable in \((a, b)\) is continuous at \(x=a\) and \(x=b\) and moreover \(f(a)=0=f(b)\). Then
- A there exists at least one point \(c\) in \((a,b)\) such that \(f^{\prime}=f\)
- B \(f^{\prime}(x)=f(x)\) does not hold at any point 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 & Solution
Correct Answer
(A) there exists at least one point \(c\) in \((a,b)\) such that \(f^{\prime}=f\)
Step-by-step Solution
Detailed explanation
Let \(g(x)=e^{-x} f(x)\) such that \(g(a)=0, g(b)=0\) and \(g(x)\) is continuous and differentiable. Then, for atleast one value of \(c \in(a, b)\) such that \(g(c)=0\) Now, \(g(x)=e^{-x} f^{\prime}(x)+\left(-e^{-x}\right) f(x)\)…
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