Question

Suppose $f$ is differentiable at $x=1$ and $\underset{h\to 0}{lim}\frac{1}{h}f(1+h)=5$, then

• A. $f^{\prime }\left(1\right)=4$
• B. $f^{\prime }\left(1\right)=3$
• C. $f^{\prime }\left(1\right)=6$
• D. None of these

Answer 1

The correct option is D None of these

The right hand derivative is given by:

$f^{\prime }\left(x\right)={lim}_{h\to 0}\frac{f(1+h)-f\left(1\right)}{h}$

Now, $f\left(1\right)=0$

$\Rightarrow f^{\prime }\left(x\right)={lim}_{h\to 0}\frac{f(1+h)-0}{h}$

$\Rightarrow f^{\prime }\left(x\right)={lim}_{h\to 0}\frac{f(1+h)}{h}$

$\Rightarrow f^{\prime }\left(x\right)=5$

So none of the options are correct. ....Answer