If $f(x) = sin\, (sin\, x)$ and $f'' (x) + tan \,x\,f'(x)+ g(x) = 0,$ then $g(x) $ is :
- $cos^2\, x\, cos\, (sin\, x)$
- $sin^2\, x\, cos\, (cos\, x)$
- $sin^2\, x\, sin\, (cos\, x)$
- $cos^2\, x\, sin\, (sin\, x)$
The Correct Option is D
Solution and Explanation
$\Rightarrow f '\left(x\right)=cos\left(sin\,x\right).cos\,x$
$\Rightarrow f ''\left(x\right)=-sin\,\left(sin\,x\right).cos^{2}\,cos\left(sin\,x\right).\left(-sin\,x\right)$
$=-cos^{2}\,x. sin\left(sin\,x\right)-sin\,x.cos\left(sin\,x\right)$
Now $f ''\left(x\right) + tan x .f '\left(x\right) + g \left(x\right) = 0$
$\Rightarrow g\left(x\right) = cos^{2}\, x . sin \left(sin\, x\right) + sin\, x . cos \left(sin\, x\right)- tan \,x . cos\, x . cos \left(sin\, x\right)$
$\Rightarrow g\left(x\right) = cos^{2}\,x . sin \left(sin\, x\right).$
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- Solutions
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.