For all twice differentiable functios

Question:

For all twice differentiable functios $f: \mathrm{R} \rightarrow \mathrm{R}$, with $f(0)=f(1)=f^{\prime}(0)=0$

  1. (1) $f^{\prime \prime}(x) \neq 0$ at every point $x \in(0,1)$

  2. (2) $f^{\prime \prime}(x)=0$, for some $x \in(0,1)$

  3. (3) $f^{\prime \prime}(0)=0$

  4. (4) $f^{\prime \prime}(x)=0$, at every point $x \in(0,1)$


Correct Option: , 2

Solution:

Let $f: \mathrm{R} \rightarrow \mathrm{R}$, with $f(0)=f(1)=0$ and $f^{\prime}(0)=0$

$\because f(x)$ is differentiable and continuous and

$f(0)=f(1)=0$

Then by Rolle's theorem, $f^{\prime}(c)=0, c \in(0,1)$

Now again

$\because f^{\prime}(c)=0, f^{\prime}(0)=0$

Then, again by Rolle's theorem,

Now again

$\because f^{\prime}(c)=0, f^{\prime}(0)=0$

Then, again by Rolle's theorem,

$f^{\prime \prime}(x)=0$ for some $x \in(0,1)$

 

Leave a comment

Close
faculty

Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now