# Real Variable. Function.

Let $f:[-1,1]\rightarrow R$ a continuous function: $-1\leq f(x)\leq 1 \forall x\in [-1,1]$. Prove that exist some $c\in[-1,1]: f(c)=c^3.$

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Homework? What did you try? –  Did Jan 21 '13 at 15:47
@RafaelJiménezGuerra your above comments means: "Hey seriously if you do not have help, do not help, but do not let unnecessary comments"...dont be rude here in MSE.no one will give answer to you, -1 for you –  La Belle Noiseuse Jan 21 '13 at 16:01
If really you are not looking for answers but for hints, why do you never indicate what you tried and do you always reproduce verbatim the text of the exercise? This conveys exactly the opposite message. –  Did Jan 21 '13 at 16:15

consider the continous(why?) function $g(x)=f(x)-x^3$ , $g(1)=f(1)-1\le 0$(why?) and $g(-1)=f(-1)+1\ge 0$(why?) hence there exist $c\in [-1,1]$ such that $g(c)=0\Rightarrow f(c)-c^3=0\Rightarrow f(c)=c^3$(why?)