I forgot how much i hated min max problems... If you need more explination, just ask me and i can explain

1. 2 numbers differ by 12. The sum of the square is a minimum. What are the two numbers?

let the 2 numbers be x and x + 12

minimum = x^2 + (x+12)^2

= 2x^2 + 24x + 144

= 2(x^2 +12) + 144

= 2(x^2 +12 + 36 - 36) + 144

= 2(x^2 +12 + 36) - 72 + 144

= 2(x+6)^2 + 72

x = -6

x+ 12 = 6

Therefore the two numbers are -6 and 6

~~I might have screwed this up, i don't think the numbers look right, but i'm tired and ill need to consult notes from two years ago, which are in my basement. ~~

Nope, second check back this makes sense, the MAX SUM is 72, which is in the equation.

2. A fence is built with one side along the river. There fence has 2 sections and is built with 120 m of fencing. What are the dimensions of the fence to achieve the max area?

Diagram:

Ok, So assume the smaller sides are x and the side opposite the river is y

y = 120 - 3x

MArea = xy

= x (120 - 3x)

= -3x^2 + 120x

=-3(x^2 - 40)

= 3(x^2 - 40 + 400 - 400)

= 3(x-20)^2 + 1600

Therefore, x = 20

y = 120 - 3(20)

y = 60

The dimensions of the fence are 20 x 60, with the max area being 1600

3. Using a formal check prove that (5+ the square root of 7)/3 is a solution of 3x squared - 10x + 16=0

To do a formal check you sub the solution for x and solve each side individually (in this case the rs is 0 so just drop it) and once you expand and simplify both side should equal each other, so in this case the rs once simplified should equal 0. Also the solution is a radical where the numerator is 5 + the square root of 7 over 3 (like a fraction) if that helps

Unless i messed up, this question is not possible. While the sqrt(7) does cancel out, the 16 screws with everything. I can try again, but can you write it out on paper so i dont misunderstand the question.