Let `f(x),xgeq0,` be a non-negative continuous function, and let `f(x)=int_0^xf(t)dt ,xgeq0,` if for some `c >0,f(x)lt=cF(x)` for all `xgeq0,` then show that `f(x)=0` for all `xgeq0.`

Explanation

Given that `F(x) = int_(0)^(x)f(t)dt`
`therefore F^(')(x) = f(x)`
Also, given that `f(x)lecF(x) AA xge0`
Also, given that `f(x) lecF(x) AA xge0`
`therefore f(0)lecF(0)=0`
`therefore f(0)le0`
But given that f(x) is non-negative function on `[0,infty]`
`therefore f(x) ge0`
`therefore f(0) ge0`
Thus, from equation (2) and (3), f(0)=0.
Again, `f(x) lecF(x) AA xge0`
or `F^(')(x) -cF(x) le0`
or `F^(')(x) - cF(x) le0 AA xge0` [using equation] or `e^(-cx)F^(')(x)-ce^(-cx)F(x)le0` [multiplying both sides by `e^(-cx)`. I.F)
or `d/(dx)[e^(-cx)F(x)]le0`
Thus,`g(x) = e^(-cx)F(x)` is a decreasing function on `[0,infty]` i.e.,
`g(x) le g(0)` for all `xge0`
But `g(0) = F(0)=0`
`therefore g(x) le0 AA xge0`
or `e^(-cx)F(x) le0 AA xge0`
or `f(x) lecF(x) le0 AA xge0[therefore cgt0` and using `f(x) lecF(x)`]
or `f(x) le0 AA xge0`
But given `f(x)ge0`
`therefore f(x)=0 AA xge0`