question_answer

A function f is continuous and differentiable on Ro and satisfies the condition \[x\,\,f'(x)+f(x)=1\] throughout its domain, with \[f(1)=2\]. Then the range of the function is

A)  \[(-\infty ,\infty )\]                      

B)  \[(-\infty ,1)\cup (1,\infty )\]

C)  \[(0,\infty )\]                 

D)  \[(1,\infty )\]

Correct Answer: B

Solution :

\[x\,f(x)\,=x+C\] \[\therefore \,\,f(1)=1+C\Rightarrow \,C=1\] \[\Rightarrow \,\,f(x)\,=\frac{x+1}{x}\] \[\therefore \,\,f(x)=1+\frac{1}{x};\,\,f'(x)\,=\frac{-1}{{{x}^{2}}}\] \[\Rightarrow \] f is always derivable and decreasing in its domain \[\Rightarrow \]monotonic Also f? is not bounded. The graph of \[y=f(x)\] is as shown           Y = 1 and x = 0 are the two asymptotes And range is R - {1}.