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- Jan 26, 2012

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For any continuous real-valued function $f$ defined on the interval $[0,1]$, let

\begin{gather*}

\mu(f) = \int_0^1 f(x)\,dx, \,

\mathrm{Var}(f) = \int_0^1 (f(x) - \mu(f))^2\,dx, \\

M(f) = \max_{0 \leq x \leq 1} \left| f(x) \right|.

\end{gather*}

Show that if $f$ and $g$ are continuous real-valued functions defined on the interval $[0,1]$, then

\[

\mathrm{Var}(fg) \leq 2 \mathrm{Var}(f) M(g)^2 + 2 \mathrm{Var}(g) M(f)^2.

\]

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