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=
just solve
-14 - b^2/-4 = 107
Y = -x^2 + Bx - 14.
x = Xv = -B/- = B/2.
-(B/2)^2 + B*B/2 - 14 = 107.
--B^2/4 + B^2/2 - 14 = 107,
-B^2/4 + 2B^2/4 - 14 = 107,
B^2/4 = 121,
B^2 = 484,
B = 22.
To find the values of "b" such that the function has a maximum value of 107, we need to use the concept of vertex form for the quadratic function.
The general form of a quadratic equation is given by:
f(x) = ax^2 + bx + c
In order to find the maximum value of the function, we need to find the vertex of the parabola. The x-coordinate of the vertex can be found using the formula:
x = -b / (2a)
In this case, the equation is f(x) = -x^2 + bx - 14, so a = -1.
Let's find the x-coordinate of the vertex:
x = -b / (2*(-1))
x = b / 2
Now, substitute this value of x into the original function to find the y-coordinate of the vertex:
f(x) = -x^2 + bx - 14
107 = -(b/2)^2 + b(b/2) - 14
Simplifying this equation leads to:
107 = -b^2 / 4 + b^2 / 2 - 14
107 = -b^2 / 4 + 2b^2 / 4 - 14
Multiplying through by 4 to get rid of the fractions:
428 = -b^2 + 2b^2 - 56
Now, combine like terms and solve the quadratic equation for b:
-b^2 + 2b^2 = 428 - 56
b^2 = 372
b = ± sqrt(372)
b ≈ ± 19.28
Therefore, the values of "b" such that the function has a maximum value of 107 are:
(smaller value): b = -19.28
(larger value): b = 19.28