1.Let $I = [a , b]$ and let $f : I \to \mathbb{R}$ be a continuous function on $I$ such that for each $x$ in $I$ there exists $y$ in $I$ such that $|f(y)| \le \frac{1}{2}|f(x)|$.
Prove there exists a point $c$ in $I$ such that $f (c) = 0$.
2.Let $f$ be continuous on the interval [$0, 1$] to $\mathbb{R}$ and such that $f (0) = f (1)$. Prove that there exists a point $c$ in [$0, 1/2$] such that $f (c) = f (c+ 1/2)$.
here are two problems on which I have completely stuck.can anyone guide me please to solve these problems
Answer
1) If $|f(x)| > 0$ for all $x \in [a,b]$ let $x_0$ be the point where it attains its minimum. Can the stated condition hold at $x_0$?
2) If $f(0) = f(1/2)$ you are done. Otherwise consider $g(x) = f(x) - f(x + 1/2)$. What can you say about $g(0)$ and $g(1/2)$? Does $g$ vanish somewhere in between?
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