Let $$3f\left( x \right) - 2f\left( {\frac{1}{x}} \right) = x,$$ then $$f'\left( 2 \right)$$ is equal to :
A.
$$\frac{2}{7}$$
B.
$$\frac{1}{2}$$
C.
$$2$$
D.
$$\frac{7}{2}$$
Answer :
$$\frac{1}{2}$$
Solution :
$$\eqalign{
& 3f\left( x \right) - 2f\left( {\frac{1}{x}} \right) = x.....(1) \cr
& {\text{Put }}x = \frac{1}{x},{\text{ then }}3f\left( {\frac{1}{x}} \right) - 2f\left( x \right) = \left( {\frac{1}{x}} \right).....(2) \cr
& {\text{Solving (1) and (2), we get}} \cr
& 5f\left( x \right) = 3x + \frac{2}{x}\, \Rightarrow f'\left( x \right) = \frac{3}{5} - \frac{2}{{5{x^2}}} \cr
& \therefore \,f'\left( 2 \right) = \frac{3}{5} - \frac{2}{{20}} = \frac{1}{2} \cr} $$
Releted MCQ Question on Calculus >> Differentiability and Differentiation
Releted Question 1
There exist a function $$f\left( x \right),$$ satisfying $$f\left( 0 \right) = 1,\,f'\left( 0 \right) = - 1,\,f\left( x \right) > 0$$ for all $$x,$$ and-
A.
$$f''\left( x \right) > 0$$ for all $$x$$
B.
$$ - 1 < f''\left( x \right) < 0$$ for all $$x$$
C.
$$ - 2 \leqslant f''\left( x \right) \leqslant - 1$$ for all $$x$$
If $$f\left( a \right) = 2,\,f'\left( a \right) = 1,\,g\left( a \right) = - 1,\,g'\left( a \right) = 2,$$ then the value of $$\mathop {\lim }\limits_{x \to a} \frac{{g\left( x \right)f\left( a \right) - g\left( a \right)f\left( x \right)}}{{x - a}}$$ is-
Let $$f:R \to R$$ be a differentiable function and $$f\left( 1 \right) = 4.$$ Then the value of $$\mathop {\lim }\limits_{x \to 1} \int\limits_4^{f\left( x \right)} {\frac{{2t}}{{x - 1}}} dt$$ is-