Question

If \[f:(-1,1)\to R\] be a differentiable function with \[f(0)=-1\] and \[f'(0)=1\] .Let \[g(x)={{[f\{2f(x)+2\}]}^{2}}\] then, \[g'(0)\] is equal to
A. \[4\]
B. \[-4\]
C. \[0\]
D. \[-2\]

Answer 1

Hint: Firstly we will differentiate the given function \[g(x)\] with respect to \[x\] on both sides and after differentiation put the value of \[x=0\] and also put the values of \[f(0)=-1\] and \[f'(0)=1\] to find the value of \[g'(0)\] and check which option is correct in the given options.

Complete step by step answer:
In mathematics function is defined as a binary relation between the two sets. A function consists of two sets of objects and a correspondence or rule that associates an object in one of the sets with an object in the other set. With the help of a function a relation can be defined between the thing of one type to the other type by some laws hence can say that a function is a relation of a particular type. Functions are also called a subset of relations.
A function consists of two nonempty sets \[X\] and \[Y\] , and a rule \[f\] that associates each element \[x\] in \[X\] with one and only one element \[y\] in \[Y\] . This is symbolized by \[f:\]\[X\to Y\] and reads the function from \[X\] into \[Y\] .
A function is a rule which maps a number to another unique number. In other words, if we start off with an input, and we apply the function, we get an output.
The input to the function is called the independent variable, and is also called the argument of the function. The output of the function is called the dependent variable.

Now according to the question:
We have given that \[g(x)={{[f\{2f(x)+2\}]}^{2}}\]
 To find the value of \[g'(0)\] we have to find \[g'(x)\]
Hence we are differentiating the function \[g(x)\] with respect to \[x\] on both sides
\[\Rightarrow \]\[\dfrac{d}{dx}g(x)=\dfrac{d}{dx}{{[f\{2f(x)+2\}]}^{2}}\]
\[\Rightarrow \]\[g'(x)=2[f\{2f(x)+2\}]\cdot f’\{2f(x)+2\}\cdot \dfrac{d}{dx}[2f(x)+2]\]
\[\Rightarrow \]\[g'(x)=2[f\{2f(x)+2\}]\cdot f'\{2f(x)+2\}\cdot 2f'(x)\]
Putting \[x=0\] to find the value of \[g'(0)\] we will get:
\[\Rightarrow \]\[g'(0)=2[f\{2f(0)+2\}]\cdot f'\{2f(0)+2\}\cdot 2f'(0)\]
We have given that \[f(0)=-1\] and \[f'(0)=1\] , putting these values in the equation:
\[\Rightarrow \]\[g'(0)=2[f\{2\times (-1)+2\}]\cdot f'\{2\times (-1)+2\}\cdot 2\times 1\]
\[\Rightarrow \]\[g'(0)=2[f\{-2+2\}]\cdot f'\{-2+2\}\cdot 2\times 1\]
\[\Rightarrow \]\[g'(0)=2\times f(0)\times f'(0)\times 2\]
Again putting the values of \[f(0)=-1\] and \[f'(0)=1\] we will get:
\[\Rightarrow \]\[g'(0)=2\times (-1)\times 1\times 2\]
\[\Rightarrow \]\[g'(0)=-4\]

So, the correct answer is “Option B”.

Note: In algebra, the word function first appeared in \[1673\] . Leibnitz and J. Bernoullie are the main contributors in the study of relations and functions. We must also remember that the inverse of a function is always unique and the inverse relation of functions is symmetric.