<s>[INST] The sum of two positive numbers is 1. The sum of their cubes is a maximum. What are the numbers? I set this up and end up finding the minimum (the two numbers would both be $1/2$).  To find a maximum value, I could reflect the functions and use $y^3-x^3$ but I still end up finding $1/2$ as the two numbers.  It does make sense that two two values to produce a max would be 0 and 1 but I can't figure out how to set up the problem from the start.  What I have is...
$x+y=1 \\
x^3 + y^3 = max$
Subsitution...
$x^3 - (1-x)^3 = max \\
x^3 - 1 + 3x - 3x^2 + x^3 = max \\
2x^3 - 3x^2 + 3x - 1 = max \\
6x^2 - 6x + 3 = 0 \\
x = 1/2 \text{ which makes }y = 1/2 \\
$
That is where my issue is.  How do I set it up to find the sum of the cubes to be a max?
 [/INST] If $x\ge 0$ and $y\ge 0$ and $x+y=1$, then $y=1-x$ and $0 \le x \le 1$
So $x^3+y^3=x^3 + (1-x)^3 = 3x^2-3x+1$. This is a parabola with axis of symmetry, and minimum, at $x = \frac 12$. It follow that the max values will be at $x=0$ and $x=1$.

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