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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

Answer»

`infty`
1
2
0

Solution :`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`


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