I need help in the following questions :

Qn1 : Let f ; g be continuous functions from [0, 1] onto [0, 1]. Prove that there is a ∈ [0, 1] such that f(g(a)) = g(f (a)).

Qn2 : Compute the limit lim(n→∞) integrate (0,pi,cos(x^n))

Thanks in advance.

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- Jan 27th 2010, 07:32 AM #1

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- Jan 2010
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## need help

I need help in the following questions :

Qn1 : Let f ; g be continuous functions from [0, 1] onto [0, 1]. Prove that there is a ∈ [0, 1] such that f(g(a)) = g(f (a)).

Qn2 : Compute the limit lim(n→∞) integrate (0,pi,cos(x^n))

Thanks in advance.

- Jan 27th 2010, 08:20 AM #2
(1)

First observe that $\displaystyle g\circ f$ and $\displaystyle f\circ g$ are onto since $\displaystyle f$ and $\displaystyle g $ are.

Suppose that $\displaystyle g(f(x))>f(g(x))$ for all $\displaystyle x\in[0,1]$. This means that $\displaystyle g(f(x_0))> 1$ for some $\displaystyle x_0\in[0,1]$. And $\displaystyle g(f(x))< f(g(x))$ for all $\displaystyle x$ implies there exists a $\displaystyle x_0\in [0,1]$ such that $\displaystyle g(f(x_0)) < 0$.

These are contradictions, hence the conclusion follows. (By continuity)

(You can use intermediate value theorem to show: $\displaystyle f(g(x))-g(f(x)) < 0$ and $\displaystyle f(g(x))-g(f(x))> 0$ for some $\displaystyle x$ implies there exists a $\displaystyle a\in [0,1]$ such that $\displaystyle f(g(a))-g(f(a)) = 0$, by continuity of $\displaystyle f(g(x))-g(f(x))$).