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Easy: Two numbers with a fixed sum
Name the quantity to maximize and write it as a formula. Use the constraint (a fixed perimeter, a fixed length of fence, a fixed sum) to write it in terms of one variable. Set its derivative equal to zero, solve, and check that the value gives a maximum, for example because the function is a downward parabola.
Use derivatives to find the maximum.
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The quantity to maximize usually depends on two variables. The constraint links them, so you can substitute and get a function of one variable to differentiate.
Check the sign of f′ on each side, use the second derivative, or compare with the values at the ends of the allowed interval.
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