The sum of the first number squared and the second number is 57 and the product is a maximum.

Asked by maham237 @ 24/12/2020 in Mathematics viewed by 350 People

Find two positive numbers that satisfy the given requirements. the sum of the first number squared and the second number is 57 and the product is a maximum.

Answered by maham237 @ 24/12/2020

We're looking for two positive numbers that satisfy this constraint:

$x^2+y=57 xy=\text{ max}$

The question is, how would we find the maximum? Well, we can solve for y to get:

$y=57-x^2$

Now, we sub our new found y value into the product to get:

$57x-x^3=\text{ max}$

So, to find the extrema of this we have to figure out when the derivative is 0, so:

$57-3x^2=0 \implies \\3x^2=57 \implies\\ x^2=19 \implies\\ x=\sqrt{19}$

That would mean our second number would be 38, and the maximum product would be $38\sqrt{19}$

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