For each continuous function $f: [0,1] \rightarrow R,$ let $I\left(f\right) = \int_0^1 x^2 f\left(x\right)\: \textrm{d}x.$ and $J\left(f\right) = \int_0^1 x \left(f\left(x\right)\right)^2 \: \textrm{d}x.$ Find the maximum value $I\left(f\right) - J\left(f\right)$ over all such functions f.
So the problem is:
Step 1. For what values of $x$ does $\frac{\textrm{d}x}{\textrm{d}y} \left[ \int_0^1 x^2f\left(x\right)\: \textrm{d}x - \int_0^1 xf\left(x\right)^2\:\textrm{d}x\right] = 0 $
Step 2. For what values of $x$ is this negative $\frac{\textrm{d}^2x}{\textrm{d}y^2} \left[ \int_0^1 x^2f\left(x\right)\: \textrm{d}x - \int_0^1 xf\left(x\right)^2\:\textrm{d}x\right] = 0 $
Not sure exactly how to do that.
Page 281 Problem #80 in Calculus 9$^{th}$ edition by Larson No, its not homework its way to difficult for class, but I like math and the last problem is the most fun and I learn the most from.