Question

1. Find all values of p such that 2(x+4)(x-2p) has a minimum value of -18. I know that y is the minimum value,I know x is -4. But, I am not sure how to find p.

2.The temperature of a point (x,y) in the plane is given by the expression x^2+ y^2 - 4x + 2y.What is the temperature of the coldest point inthe plane? I know the minimum point is the coldest point on the plane. When I try to factor it I get x(x-4)+ y(y+2) but I do not know what to do from there.

Answer 1

You know that the minimum value is midway between the roots, so it occurs at

x = ((-4)+(2p))/2 = p-2

So, we know that

2(p-2+4)(p-2-2p) = 2(p+2)(-2-p)
= -2(p+2)^2 = -18
So, p+2 = 3 or -3
So, p = 1 or -5
Check:

2(x+4)(x-2) has its minimum at x = -1
2(3)(-3) = -18

2(x+4)(x+10) has its minimum at x = -7
2(-3)(3) = -18