WBJEE · Maths · Application of Derivatives
Let \(f\) be any continuously differentiable function on \([a, b]\) and twice differentiable on \((a, b)\) such that \(f(a)=f^{\prime}(a)=0\) and \(f(b)=0\) Then.
- A \(f^{\prime \prime}(a)=0\)
- B \(f^{\prime}(x)=0\) for some \(x \in(a, b)\)
- C \(f^{\prime \prime}(x) \neq 0\) for some \(x \in(a, b)\)
- D \(f^{\prime \prime \prime}(x)=0\) for some \(x \in(a, b)\)
Answer & Solution
Correct Answer
(C) \(f^{\prime \prime}(x) \neq 0\) for some \(x \in(a, b)\)
Step-by-step Solution
Detailed explanation
We have, \(f\) is continuous and differentiably function on \([a, b]\) Also, \(f(a)=f(b)=0\) By Rolle's theorem, there exists \(c \in(a, b)\) such the \(f^{\prime}(c)=0\) Thus, there exists \(x \in(a, b)\) such that \(f^{\prime}(x)=0\) Let at \(x=c \in(a, b), f^{\prime}(c)=0\)…
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