extreme values of a continuous function

[dbob85]

Find two nonnegative numbers whose sum is 8 and the product of whose squares is as large as possible.

I don't know how in the world you set this one up. All I have so far is "x+y=8"

 

[stapel]

What would be the squares of your numbers? What would be the product of the squares?

Solve the equation "x + y = 8" for one of the variables. (It doesn't matter which.) This gives you a variable for the one number and an expression for the other number.

Square the variable.

Square the expression.

"Product" means "multiply", so multiply the two squares.

Maximize the resulting expression.

Eliz.

 

[dbob85]

ok so when i get x^4-16x^3+64x^2 do i just take the derivative of it and then i'm lost again

 

[stapel]

get x^4-16x^3+64x^2 [when I multiply the two expression. Then] do just take the derivative of it[? And, if so, then I'm] lost again[.]

Has your class not covered how to use the derivative to find max/min points...?

Thank you.

Eliz.

 

[galactus]

You shouldn't have to square them. I think there's an easier way.
When you maximize or minimize a distance, there is a trick based on the observation that the distance and the square of the distance have their max's and min's at the same point.

We can use this trick here. Why not?.

\(\displaystyle x+y=8\)....[1]

\(\displaystyle P=xy\).....[2]

Solve [1] for y, sub into [2], differentiate, set to 0 and solve for x.