Let `f(x)` be a non-positive continuous function and `F(x)=int_(0)^(x)f(t)dt AA x ge0` and `f(x) ge cF(x)` where `c lt 0` and let `g:[0, infty) to R` be a function such that `(dg(x))/(dx) lt g(x) AA x gt 0` and `g(0)=0`
The total number of root(s) of the equation `f(x)=g(x)` is/ are

Options

A. `infty`
B. 1
C. 2
D. 0

Answer

B

Explanation

`f(x) le0` and `F^(')(x)=f(x)`
Now, `f(x) ge cF(x)`
or `e^(-cx)F^(')(x) -ce^(-cx)F(x) ge0`
Thus, `e^(-cx)F(x)` is an increasing function
`therefore e^(-cx) F(x) ge e^(-c(0))F(0)`
or `e^(-cx)F(x) ge`
or `F(x) ge0` [as `f(x) ge cF(x)` and c is positive.
Also, `(dg(x))/(dx) lt g (x) AA x gt 0`
or `(d/(dx)) e^(-x)g(x) lt 0`
Thus, `e^(-x)g(x)` has one solution, `x=0`