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Find weight functions so that the weighted average has given values

Let f(x) = x^2 for 0 \leq x \leq 1. We know the average of f on [0,1] is \frac{1}{3}. Find a nonnegative function w such that the weighted average of f on [0,1] is

  1. \frac{1}{2},
  2. \frac{3}{5},
  3. \frac{2}{3},

First, we recall that given functions f and w (with w nonnegative) defined on an interval [a,b], the weighted average is given by

    \[ A(f) = \frac{\int_a^b w(x) f(x) \, dx }{\int_a^b w(x) \, dx}. \]

  1. Let w(x) = x, then

        \[ A(f) = \frac{\int_0^1 x \cdot x^2 \, dx}{\int_0^1 x \, dx} = \frac{1/4}{1/2} = \frac{1}{2}. \]

  2. Let w(x) = x^2, then

        \[ A(f) = \frac{\int_0^1 x^2 \cdot x^2 \, dx}{\int_0^1 x^2 \, dx} = \frac{1/5}{1/3} = \frac{3}{5}. \]

  3. Let w(x) = x^3, then

        \[ A(f) = \frac{\int_0^1 x^3 \cdot x^2 \, dx}{\int_0^1 x^3 \, dx} = \frac{1/6}{1/4} = \frac{2}{3}. \]

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