Question

Let $y=f(x)$ be a function such that $\frac{d y}{d x}=y-2 x+2, f(0)=1$. The equation $f(x)=k x(k \in$ R) has $n$ distinct real roots then which of the following is/are correct?

• A. if $k=e+2$ then $n=1$
• B. $ if $k > e +2$ then $n =2$
• C. if $2< k< e+2$ then $n=0$
• D. if $k< 2$ then $n=1$

Answer 1

Answer: (A), (B), (C), (D)

$ y ^4+(-1) y =2-2 x $
$\Rightarrow ye ^{- x }=\int(2-2 x ) e ^{- x } dx$
$f ( x )=2 x + ce ^{ x } \text { but } f (0)=1$
$\Rightarrow f ( x )=2 x + e ^{ x }$
$\frac{ e ^{ x }}{ x }= k -2$
Now $g(x)=\frac{e^x}{x}$ now interpret from the graph.