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## Homework Statement

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

[tex]M_1^2\leq 4 M_0 M_2[/tex]

## 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:

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

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.

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