Let-m-n-be-given-positive-integers-If-x-amp-y-are-positive-numbers-such-that-x-y-S-S-is-a-constant-find-the-value-of-x-and-y-that-maximize-Q-x-m-y-n-

Question Number 140734 by liberty last updated on 12/May/21

Let m, n be given positive integers .

If x & y are positive numbers such that

x + y = S, S is a constant, find

the value of x and y that maximize

Q = x^{m} y^{n} .

Commented by mr W last updated on 12/May/21

i f m a n d n b o t h a r e e v e n, t h e r e i s

m i n i m u m .

i f o n l y o n e o f m, n i s e v e n, t h e r e i s n o

m a x i m u m a n d n o m i n i m u m .

i f b o t h m a n d n a r e o d d, t h e r e i s

m a x i m u m .

Answered by EDWIN88 last updated on 12/May/21

setting \frac{dQ}{dx} = {mx}^{m - 1} {(S - x)}^{n} - n {(S - x)}^{n - 1} x^{m} = 0

we obtain x = \frac{mS}{m + n} . The first - derivative test

show that this yields a relative maximum

and therefore by the uniqueness of critical

number, an absolute maximum when x = \frac{mS}{m + n}, y = \frac{nS}{m + n}

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