Show that if the sum of two positive numbers is given, their product is greatest when they are equal and if the product of two positive numbers is given, their sum is least when they are equal.

Answer 1

The sum of the variables in the second part cannot be minimized

without calculus, as the sum function is NOT quadratic but hyperbolic.

X+Y = S > 0 is the sum

The product is P = X*Y = X*(S-x) = sx - x^2 = -x^2 + sx

A=-1 B=s C = 0

Extrema occurs at -B/(2a) = -s/(2*-1) = -s/-2 = s/2

So when x=s/2 then Y = S - s/2 = (2s-s)/2 = s/2.

Therefore, the product is maximized when x=y=s/2 <---- they are both half the sum.

X*Y = p >0 is the product

The sum is S = x+y = x + p/x = (x^2 + p)/x

The extrema occur when the derivative is zero:

0=dS/dx = { x(2x) - (x^2 + p)} / x^2

= { 2x^2 - x^2 - p }/ x^2

= {x^2 - p} / x^2

So the extrema occur when x^2 = p ---> x = sqrt(p) since all values are positive

Since x*y = p

sqrt(p)* y = p

y = sqrt(p)

So the sum is at it's extrema when x=y=sqrt(p)