Typical calculus qn. try to solve without calculus

[Legendon]

Homework Statement

The sum of two non-negative numbers is 20. Find the numbers (a) if the sum of their squares is to be as large
as possible; (b) if the product of the square of one number
and the cube of the other is as large as
possible; (c) if one number plus the square root of the
other is as large as possible.

Homework Equations

The Attempt at a Solution

a.
(x+y)(x+y)=x^2 +y^2 +2xy.
x^2 +y^2=(x+y)^2 -2xy=400-2xy<=400
Maximum would be when 2xy=0, x=0, y=20.
So is this fine or clear enough? Any better ideas?
b.
no idea. i can do it by calculus but why did my prof check at the endpoints ?
c.
no idea. i can do it by calculus but why did my prof check at the endpoints ?

 

[Mark44]

Homework Statement

The sum of two non-negative numbers is 20. Find the numbers (a) if the sum of their squares is to be as large
as possible; (b) if the product of the square of one number
and the cube of the other is as large as
possible; (c) if one number plus the square root of the
other is as large as possible.

Homework Equations

The Attempt at a Solution

a.
(x+y)(x+y)=x^2 +y^2 +2xy.

This is wrong.
For one thing, the problem is asking about how to make the sum of the squares as large as possible. What you have is the square of the sum, which is different.

For another, you have ignored the given information that the two numbers add up to 20.

x^2 +y^2=(x+y)^2 -2xy=400-2xy<=400
Maximum would be when 2xy=0, x=0, y=20.
So is this fine or clear enough? Any better ideas?
b.
no idea. i can do it by calculus but why did my prof check at the endpoints ?
c.
no idea. i can do it by calculus but why did my prof check at the endpoints ?

A maximum or minimum of a function f can come at any of three places:
1) At a point where f'(x) = 0
2) At an endpoint of the domain (if f has domain restrictions)
3) At a point in the domain at which f'(x) is undefined

 

[HallsofIvy]

Each of those problems can be done without calculus because they all reduce to quadratic equations. And you can find the maximum or minimum by completing the square.

 

[Hammie]

so, by the constraint you have two numbers which add up to twenty. Call one x, and the other 20-x. square both, and add.

what do you know about a quadratic that gives a maximum or minimum?