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

Question

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

Answer 1

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.