For a continuous and nonnegative function f on , let .
We will show more general statement:
Let f be continuous, nonnegative function and g be continuous, positive function on [0, 1] and . Show that
If for all , the inequality (1) is hold.
If that mean there is such that >0(note that f is nonnegative). On the other hand, one has f, g are continuous on [0, 1] and g is positive so >0 for all and >0.
Then the inequality is equivalent to:
Let , we will prove that is increasing,
It is easy to see it, by the inequality Cauchy-Schwarz for integral:
(It is our claim.)
From the fact is increasing, we get:
, that mean:
(we have done)
Now the problem of C.lupu, we only need to choose g(x)=x (One has x>0 for all -that is sufficent. And g(x)=x is obviously continuous on [0, 1])
Let be a real-valued continuously differentiable function on $[a, b]$ with positive derivative on $(a, b)$. Prove that, for all pairs with and , there exists such that
Because , and are the same sign. Without lost generality, we can assume that and are positive (Otherwise, we consider -f). Then for all , by Lagrange’s Theorem we get , hence .
Now, consider two function: and are continuously differentiable on , by Cauchy’s Theorem there is