1. ## Maximisation and Minimisation Q Help please!

Hey guys, how do i do the following question?

Show that if the sum of 2 positive integers is given, their product is greatest when they are equal and if the product of the 2 positive integers is given, their sum is least when they are equal?

Thanks

2. ## Re: Maximisation and Minimisation Q Help please!

$\noindent we can write the sum of two positive integers as x+y=k, where x,y,k are positive integers \\ \Rightarrow y= k -x \qquad (1) \\ \\ now the product, P, is equal to xy \\ P=xy = x(k-x) =x^2 -kx \qquad from eqn 1 \\ \frac{dP}{dx}=2x - k \\ \therefore max at x = \frac{k}{2} \qquad since \frac{d^2P}{dx^2}=2>0. \\ \therefore y=k-\frac{k}{2}=\frac{k}{2}=x \\ \\ Hence the product is the greatest when the two numbers are equal. By a similar method, you can solve the other part$

3. ## Re: Maximisation and Minimisation Q Help please!

You can also use a symmetry argument to infer (not prove) the answer. Note that if you interchange x,y, the equations remain unchanged. This means that you'll get the same result regardless of whether you solve for x or y, which means that x and y should be equal.

However, this doesn't prove whether it's a max/min (you'll need actual calculus for that). Nonetheless, it's a quick (and elegant) qualitative method to check your results.

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