Proving M_1^2 ≤ 4M_0M_2 in Real Analysis

[ehrenfest]

Homework Statement

Suppose a \in \mathbb{R}, f is a twice-differentiable real function on (a, \infinty) and M_0,M_1,M_2 are the least upper bound of |f(x)|,|f'(x)|,|f''(x)|, respectively on (a,\infinity). Prove that

M_1^2\leq 4 M_0 M_2

Homework Equations

The Attempt at a Solution

That is equivalent to showing that M_2 x^2 +M_1 x +M_0=0 has a real solution.

I was trying to use Taylor's Theorem which says that if \alpha and \beta are distinct points in (a,\infinity) then there exists x between \alpha and \beta that makes the following equation true:

f(\beta) = f(\alpha) + f'(\alpha)(\beta-\alpha) + f''(x) (\beta-\alpha)^2/2

I could take the absolute value of both sides and then use triangle inequality but I did not see how to get anywhere with that.

 

[eok20]

Did you look at the hint given in rudin?

 

[ehrenfest]

How was I supposed to know there was a hint on the next page? That is a serious question.

 

[ehrenfest]

OK. The hint says:

If h>0, Taylor's theorem show that

f'(x) = \frac{1}{2h}[f(x+2h)-f(x)]-hf''(\xi)

for some \xi \in (x,x+2h). Hence

|f'(x)| \leq hM_2+M_0/h

I don't see how the last inequality is useful (even if you square it).

 

[eok20]

Square it and remember that it holds for ANY h > 0. There is a particular choice of h that will yield the result.