WBJEE · Maths · Differentiation
Let \(f: R \rightarrow R\) be twice continuously differentiable. Let \(f(0)=f(1)=f^{\prime}(0)=0 .\) Then,
- A \(f^{\prime \prime}(x) \neq 0\) for all \(x\)
- B \(f^{\prime \prime}(c)=0\) for some \(c \in R\)
- C \(f^{\prime \prime}(x) \neq 0\) if \(x \neq 0\)
- D \(f^{\prime}(x)>0\) for all \(x\)
Answer & Solution
Correct Answer
(B) \(f^{\prime \prime}(c)=0\) for some \(c \in R\)
Step-by-step Solution
Detailed explanation
Let function \(f(x)=x^{2}(x-1)\) \(\Rightarrow \quad f^{\prime}(x)=3 x^{2}-2 x\) and \(\quad f^{\prime \prime}(x)=6 x-2\) \(\begin{array}{ll}\text { Now, } & f(0)=f(1)=f^{\prime}(0)=0\end{array}\) Then, according to question, at \(x=\frac{1}{3}, \quad f^{\prime \prime}(x)=0\)…
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