Let $f:R\to R$ be a differentiable function at $c\in R$ and f( c ) = 0. If g(x) = If(x)I, then at x = c, g is
(a) differentiable if f’(c) = 0
(b) not differentiable
(c) differentiable if $f'(c)\ne 0$
(d) not differentiable if f’(c) = 0

Hint: To solve this question, we will first use the definition of differentiability, to find the value of derivative of function g at x = c. then, again we will modify the formula of differentiability for h = 0. And then we will use the concept of differentiability of modulus function. Then, we will discard the options and choose the right option.

Complete step-by-step answer:
Now, in question it is given that $f:R\to R$ be differentiable function at $c\in R$ and f( c ) = 0,
 and g(x) = If(x)I.
then the function g at x = c will be
$g'(c)=\displaystyle \lim_{h \to 0}\dfrac{g(c+h)-g(c)}{h}$ , where h is greater than 0, that is h > 0
Here we have g(x) = I f(x) I
So, $g'(c)=\displaystyle \lim_{h \to 0}\dfrac{\left| f(c+h) \right|-\left| f(c) \right|}{h}$
As, it is given that f(c) = 0, so $\left| f(c) \right|=0$
Then, $g'(c)=\displaystyle \lim_{h \to 0}\dfrac{\left| f(c+h) \right|-0}{h}$
$g'(c)=\displaystyle \lim_{h \to 0}\dfrac{\left| f(c+h) \right|}{h}$
Also, we can write g’(c) as
$g'(c)=\displaystyle \lim_{h \to 0}\left| \dfrac{f(c+h)-f(c)}{h} \right|$, where h > 0
$g'(c)=\left| \displaystyle \lim_{h \to 0}\dfrac{f(c+h)-f(c)}{h} \right|$
As, $\displaystyle \lim_{h \to 0}\dfrac{f(c+h)-f(c)}{h}$ denotes f’(c)
So, we can say that $\left| \displaystyle \lim_{h \to 0}\dfrac{f(c+h)-f(c)}{h} \right|=\left| f'(c) \right|$
$g'(c)=\left| f'(c) \right|$
Or, f is differentiable at x = c
Now, if f’(c) = 0 then g(x) is differentiable at x = c otherwise, left hand derivative say ( LHD )at x = c and right hand derivative ( RHD ) at x = c is different.
Now, also we know that modulus function $\left| x \right|$ is differentiable everywhere except at x = 0.
So, we can say that g is differentiable at $f'(c)\ne 0$, as we proved above that g is differentiable and equals to $g'(c)=\left| f'(c) \right|$.

So, the correct answer is “Option c”.

Note: Always remember that if we have to differentiate a function f(x) at point x = a, then we can evaluate the differentiation of f(x) by formula $f'(a)=\displaystyle \lim_{h \to 0}\dfrac{f(a+h)-f(a)}{h}$. Also, remember that function $\left| x \right|$ is differentiable everywhere except at x = 0. Try to choose the correct option by discarding wrong options first.