Proofs of inequalities using M.V.T

The mean value Theorem M. V. T

[davidbrisbane]

Good solution. I'm posit and alternative approach to showing (y - x)/2√y < √y - √x, where 0 < x < y

Assume by way of contradition that
(y - x)/2√y ≥ √y - √x, where 0 < x < y

So (y - x)/(√y - √x) ≥ 2√y,
as √y - √x > 0 and 2√y > 0
So (√y + √x) ≥ 2√y,
as (√y + √x)(√y - √x) = (y - x)
So √y + √x ≥ 2√y = √y + √y
So √x ≥ √y
So x ≥ y, which contradicts x < y.

So, we conclude
(y - x)/2√y < √y - √x,
where 0 < x < y

[fea]

How did you choose f(x) to be the square root of x?

[vladimirputinii]

what happen between 5.43 - 6.03 ie why you change the sign of the inequality

[souvikhazra]

Can you please say why the interval will be from (0, a)