Question

Find two negative numbers, and , whose product is and whose sum is a maximum
@MIT 18.02 Multiva…

Answer 1

the product is 1107, sorry typo

Answer 2

We are trying to find a maximum of the function f(x,y) = x + y subject to the condition that xy=1107 (and x<0, y<0).

Since the condition implies that y = 1107/x we can rewrite the function as f(x) = x + 1107/x

A possible maximum will occur only when the derivative of function f with respect to x is equal to zero, i.e.
\[f \prime(x) = 1-1107x ^{-2}=0\]This equation is satisfied when
\[x=-\sqrt{1107}, x=\sqrt{1107}\]Since we are only interested in negative numbers, the first root is our possible maximum.

To prove that it is a maximum (rather than a minimum) we take the second derivative of f which gives
\[f \prime \prime(x) = 2214x ^{-3}\]Substituting our value of x gives a negative value for the second derivative which proves that that value of x leads to a maximum value of the function. Note that this is simply a local maximum since for positive numbers (which we have to exclude) f(x) tends to infinity as x tends to infinity.

Substituting x back into our condition gives
\[y=-\sqrt{1107}\]