Optimization

[sly oyster]

The sum of the 1st number squared and the 2nd is 27 and the product is a maximum

[wintry lark]

x² + y = 27
y = 27 - x²
P(x) = x(27 - x²)
max -> P'(x) = 0
solve for x and y

[sly oyster]

wait wouldn't it be x^2+27?

[sly oyster]

for p(x)=x(27-x^2) would i need to foil or no

okay so i foul the p(x)=x(27-x^2) which is p(x)=27x-x^3 and then did the derivative which is p'(x)=27-3x^2

and then

i = to 0

at the end

which then i got 3

so x=3

and now idk what to do withy

[wintry lark]

x = ±3

[sly oyster]

with y

oh

then that's my answer?

[wintry lark]

then now you find the 2nd derivative

[sly oyster]

of the first derivative i did?

[wintry lark]

then plug in x = 3 and x = -3

[sly oyster]

of this p'(x)=27-3x^2

so i find the 2nd derivative of p'(x)=27-3x^2 and then plug in the 3 and -3

[wintry lark]

yes take the derivative of it

[sly oyster]

okay

thank you

[wintry lark]

no not yet

[sly oyster]

oh 😭

would p'(x)=27-3x^2 be -6x

[wintry lark]

then after that you plug in these to that 2nd derivative and choose which one is negative

[sly oyster]

so then -6(3)

okay

[wintry lark]

yes

the one which gives negative is max

and vice versa

[sly oyster]

okay so

[wintry lark]

then you plug in x = 3 to get y

[sly oyster]

so y is +-18

[wintry lark]

y = 27 - (3)² = 18

no need ±

[sly oyster]

okay

but don't i need to find the max tho?

okay x=3, y=18

and 27*2=54 and 54=max product

[wintry lark]

what's the question asking you to find?

[sly oyster]

The sum of the 1st number squared and the 2nd is 27 and the product is a maximum

[wintry lark]

are you ask to find the numbers?

[sly oyster]

yes

i mean idk what else it would be asking

thanks for the help i got it right