To find the values of b such that the function f(x) = x² + bx + 14 has a maximum value of 86, we start by observing the structure of the function.
This function is a quadratic equation in the standard form f(x) = ax² + bx + c, where a = 1, b = b, and c = 14. Since the coefficient of x² (a) is positive, this function opens upwards, which implies it has a minimum value, not a maximum. However, it appears that you’re looking for the condition under which it can be manipulated to attain a specific maximum output, which typically requires the vertex form of the parabola.
The vertex of a quadratic function f(x) = ax² + bx + c is given by the formula x = -b/(2a). In our case, we can plug in the value of a:
x = -b/(2*1) = -b/2
Next, we can find the y-coordinate of the vertex, which gives us the minimum value of the function:
f(-b/2) = (-b/2)² + b(-b/2) + 14
Substituting:
f(-b/2) = (b²/4) – (b²/2) + 14 = -b²/4 + 14
To find b such that this minimum value is equal to 86, we set:
-b²/4 + 14 = 86
Now, let’s solve for b:
-b²/4 + 14 = 86
-b²/4 = 86 – 14
-b²/4 = 72
Multiplying both sides by -4 to eliminate the fraction, we get:
b² = -288
Since the value of b² cannot be negative, this indicates no real values of b exist that allow the function to achieve the value of 86 as a maximum. Therefore, there are no real solutions for b in this case.
In conclusion, there are no values of b that will make the maximum value of the function f(x) = x² + bx + 14 equal to 86, as the function only has a minimum value.