Q109 / 155 · WBJEE 2017 · Mathematics

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 $x$
  • Correct answer
    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 $x$

Solution

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 $x=\frac{1}{3}, \quad f^{\prime \prime}(x)=0$ (i.e. $c=1 / 3$ for some $C \in R$ )

Hence, the correct answer is option: 'B'.

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