$h:[0,1] \to\mathbb{R}$ continuous, and ivt

Let $d(w)=h(w)- \bigg(\frac{w+1}{2}h(0)+\frac{2w+2}{9}h(\frac{1}{2})+\frac{w+1}{12}h(1) \bigg)$.

Then,

$$ 18d(0)+8d(\frac{1}{2})+3d(1)=0 $$

So the numbers $d(0),d(\frac{1}{2}),d(1)$, if all nonzero, are not all of the same sign. Can you finish from here ?


Let $f(x)=\frac {h(x)}{(x+1)}$ which is continuous on $[0,1]$ and $T=1/2f(0)+1/3f(1/2)+1/6f(1)$.

Then we want to prove ,$$f(w)=T \ \ \text{for some } \ w\in [0,1]$$

Let $M$ and $m$ be maximum and minimum value taken by $f(x)$.

Then note that, $m\leq T\leq M$.

Tags:

Analysis

Calculus

Continuity

Real Analysis

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