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Integral Sandwich

Age 16 to 18
Challenge Level

(i) Suppose that $f(0)=0$ and that, for $x\neq 0$, $$0 \leq {f(x)\over x} \leq 1$$ Show that $$-{1\over 2} \leq \int_{-1}^1 f(x)\,dx \leq {1\over 2}$$ (ii) Suppose that $f(0)=0$ and that, for $x\neq 0$, $$0 \leq {f(x)\over x^2} \leq 1$$ Show that $$0 \leq \int_{-1}^1 f(x)\,dx \leq {2\over 3}$$ (iii) Generalize (i) and (ii) to the case where $f(0)=0$ and, for $x\neq 0$, $$0 \leq {f(x)\over x^n} \leq 1$$ where $n$ is a positive integer.