JEE Mains · Maths · STD 12 - 5. continuity and differentiation
For all twice differentiable functions \(f: R \rightarrow R,\) with \(f(0)=f(1)=f^{\prime}(0)=0\)
- A \(f^{\prime \prime}(x)=0,\) for some \(x \in(0,1)\)
- B \(f^{\prime \prime}(0)=0\)
- C \(f^{\prime \prime}( x ) \neq 0\) at every point \(x \in(0,1)\)
- D \(f^{\prime \prime}(x)=0\) at every point \(x \in(0,1)\)
Answer & Solution
Correct Answer
(A) \(f^{\prime \prime}(x)=0,\) for some \(x \in(0,1)\)
Step-by-step Solution
Detailed explanation
\(f(0)=f(1)=f^{\prime}(0)=0\) Apply Rolles theorem on \(y=f(x)\) in \(x \in[0,1]\) \(f(0)=f(1)=0\) \(\Rightarrow f^{\prime}(\alpha)=0\) where \(\alpha \in(0,1)\) Now apply Rolles theorem on \(y =f^{\prime}( x )\) \(\operatorname{in} x \in[0, \alpha]\)…
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