For $ x \epsilon R , f (x) = | \log 2 - \sin x|$ and $g(x) = f(f(x))$, then :
- $g$ is not differentiable at $x = 0$
- $g'(0)$ = $cos(log2)$
- $g'(0)$ = $-cos(log2)$
- $g$ is differentiable at $x = 0$ and $g'(0) = -sin(log2)$
The Correct Option is B
Solution and Explanation
At $x=0, g(x)=\log _{e}(2)-\sin \left(\log _{e} 2-\sin x\right)$
$\therefore g'(x)=\cos \left(\log _{e}(2)-\sin x\right) \times \cos (x)$
$\Rightarrow g'(0)=\cos \left(\log _{e}(2)\right)$
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Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.