## Twitter proof: maximising the product with a fixed sum

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Let's prove that if you want to maximise $$ab$$ with $$a + b$$ equal to a constant value $$k$$, then you want $$a = b = \frac{k}{2}$$.

Take $$s = k/2$$. If $$a = s+h$$ then $$b = s-h$$, from which we get that $$ab = (s+h)(s-h) = s^2 - h^2$$. Because we know $$h^2 \geq 0$$, $$ab$$ is maximised when $$h = 0$$, that is $$a = b = s = k/2$$.