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.
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
1. To find the values of p such that the expression has a minimum value of -18, we can start by expanding and simplifying the expression 2(x+4)(x-2p).
First, distribute the 2 to both terms inside the parentheses:
2(x+4)(x-2p) = 2(x^2 - 2px + 4x - 8p)
Next, combine like terms:
= 2(x^2 + (4-2p)x - 8p)
To find the minimum value, we need to determine the vertex of the quadratic expression. The vertex is given by the formula (h, k), where h = -b/2a and k is the minimum value.
Comparing the expression to the standard form ax^2 + bx + c, we have a = 2, b = (4 - 2p), and c = -8p.
Using the formula for the x-coordinate of the vertex, we can substitute the values:
h = -(4 - 2p) / (2 * 2) = -(4 - 2p)/4 = -1 + p/2.
Now, we know that x = -4 at the minimum point, so we can set x = -4 and solve for p:
-4 = -1 + p/2
Multiply through by 2 to get rid of the fraction:
-8 = 2p - 2
2p = -6
p = -3.
Therefore, when p = -3, the expression 2(x+4)(x-2p) has a minimum value of -18.
2. To find the coldest point on the plane represented by the expression x^2 + y^2 - 4x + 2y, we need to find the minimum value of the expression.
First, let's group the terms containing x and y:
x^2 - 4x + y^2 + 2y
Next, complete the square for both the x and y terms.
For the x terms:
x^2 - 4x = (x^2 - 4x + 4) - 4 = (x - 2)^2 - 4.
For the y terms:
y^2 + 2y = (y^2 + 2y + 1) - 1 = (y + 1)^2 - 1.
Now, we can rewrite the expression as:
(x - 2)^2 - 4 + (y + 1)^2 - 1.
Simplifying further, we get:
(x - 2)^2 + (y + 1)^2 - 5.
From the standard form of a circle equation, we know that the minimum value of the expression (x - 2)^2 + (y + 1)^2 is 0 (when the center of the circle is the minimum point).
Therefore, the minimum value of the expression x^2 + y^2 - 4x + 2y is -5.
Hence, the temperature of the coldest point in the plane is -5.