SOLUTION: Find the value/s of k so that the minimum value of f(x) = x^2 + kx + 3 is the same as the maximum value of g(x) = k + 4x - x^2.

Algebra ->  Functions -> SOLUTION: Find the value/s of k so that the minimum value of f(x) = x^2 + kx + 3 is the same as the maximum value of g(x) = k + 4x - x^2.      Log On


   

Question 1002043: Find the value/s of k so that the minimum value of f(x) = x^2 + kx + 3 is the same as the maximum value of g(x) = k + 4x - x^2.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
your two equations are:

x^2 + kx + 3 and -x^2 + 4x + k

you want to find the max value of x^2 + kx + 3 and the min value of -x^2 + 4x + k and then set them equal to each other and then solve for k.

since these are in standard quadratic form, look for x = -b/2a in each.

x^2 + kx + 3 gets you:
a = 1
b = k
c = 3

x = -b/2a = -k/2

-x^2 + 4x + k gets you:
a = -1
b = 4
c = k

x = -b/2a gets you x = -4/-2 = 2

the min value of y for x^2 + kx + 3 will be at f(-b/2a) = f(-k/2).

the max value of y for -x^2 + 4x + k will be at f(-b/2a) = f(2).

f(-k/2) for x^2 + kx + 3 becomes (-k/2)^2 + k(-k/2) + 3.
simplify this equation to get:
f(-k/2) = -k^2/4 + 3

f(2) for -x^2 + 4x + k becomes -(2^2) + 4(2) + k.
simplify this equation to get:
f(2) = k + 4

f(-k/2) is the min value of x^2 + kx + 3
f(2) is the max value of -x^2 + 4x + k

set them equal to each other and you get:
f(-k/2) = f(2) which becomes:
-k^2/4 + 3 = k + 4

add k^2/4 to both sides of the equation and subtract 3 from both sides of the equation to get:

0 = k^2/4 + k + 4 - 3
simplify to get:
0 = k^2/4 + k + 1
multiply both sides of this equation by 4 to get:
0 = k^2 + 4k + 4

factor k^2 + 4k + 4 to get (k+2)*(k+2) = 0
solve for k to get k = -2.

the min point of x^2 + kx + 3 and the max point of -x^2 + 4x + k should be the same when k = -2.

replace k with -2 in x^2 + kx + 3 to get x^2 -2x + 3.

replace k with -2 in -x^2 + 4x + k to get -x^2 + 4x - 2.

min point of x^2 - 2x +3 is at (-b/2a,f(-b/2a))
-b/2a = 2/2 = 1
f(-b/2a) = f(1) = 1 - 2 + 3 = 2.

max point of -x^2 + 4x - 2 is at (-b/2a,f(-b/2a))
-b/2a = -4/-2 = 2
f(2) = -4 + 8 - 2 = 2.

min point of x^2 - 2x + 3 is at (1,2)
max point of -x^2 + 4x - 2 is at (2,2)

the y values are the same so the min value and max value are the same.

the graph of both equations is shown below:

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