Find the values of b such that the function has the given maximum value.

f(x) = −x^2 + bx − 14; Maximum value: 107

(smaller value): b=
(larger value): b=

oobleck said "just solve -14 - b^2/-4 = 107"
but i really don't know how to solve what oobleck gave me

you know that the vertex of y=ax^2+bx+c occurs at x = -b/2a (not the same b as in your polynomial)

So, for your polynomial x = -b/-2 = b/2
At that value, f(x) = -(b/2)^2 + (b/2)b - 14 = 107
So, now you just have to solve
-b^2/4 + b^2/2 = 121
b^2/4 = 121
b^2 = 4*121 = (2*11)^2 = 22^2
b = ±22

Note that that is exactly the same as using the vertex value I gave, which in this case is
-14 - b^2/-4 = -14 + b^2/4
Set that equal to 107 and you again have
b^2/4 = 121
b = ±22

I think the confusion is increased by the fact that "b" is used as a coefficient in your polynomial.

To find the values of b for which the given function f(x) has a maximum value of 107, we need to solve the equation -x^2 + bx - 14 = 107. It seems like there was a misunderstanding regarding the equation oobleck provided.

Let's set up the equation correctly and solve for b:

First, rearrange the equation by subtracting 107 from both sides:
-x^2 + bx - 14 - 107 = 0

Next, combine like terms:
-x^2 + bx - 121 = 0

To find the values of b, we can use the quadratic formula. Given a quadratic equation in the form ax^2 + bx + c = 0, the quadratic formula states that the solutions for x can be found using:

x = (-b ± √(b^2 - 4ac)) / 2a

In our case, the quadratic equation is -x^2 + bx - 121 = 0, which means a = -1, b = b, and c = -121.

Now, let's apply the quadratic formula to find the values of b:

x = (-b ± √(b^2 - 4ac)) / 2a
x = (-b ± √(b^2 - 4(-1)(-121))) / 2(-1)
x = (-b ± √(b^2 + 484)) / -2

To simplify this further, we can factor out -1 from the square root term:
x = (-b ± √((-1)(b^2 + 484))) / -2
x = (-b ± √(-1) √(b^2 + 484)) / -2
x = (-b ± i√(b^2 + 484)) / -2

Since we're looking for real values of b that satisfy the given maximum value condition, the discriminant (b^2 + 484) should be equal to zero to eliminate the complex part. Thus,

b^2 + 484 = 0

Subtracting 484 from both sides, we have:
b^2 = -484

Taking the square root of both sides:
√(b^2) = ±√(-484)
b = ±√(-1) √(484)
b = ± i√(484)

Since the values under the square root are negative (-1 and 484), there are no real solutions for b that satisfy the condition of a maximum value of 107 for the given function f(x) = -x^2 + bx - 14.