Find the values of b such that the function has the given maximum value. f(x) = -x² + bx - 14; Maximum value: 86

Solution:

Given,

f(x) = -x² + bx - 14;

ymax = 86

ymax is the vertex of the parabola

We know that the coordinates of the vertex of the parabola f(x) = ax² + bx + c

V(h,k) is given by V(-b/2a, -(b - 4ac)/2a)

From the given equation

a = -1, b = b, c = -14

Substituting these values

-[b² - 4(-1) (-14)]/ 4(-1) = 86

By further calculation

(b² - 56)/4 = 86

By cross multiplication

(b² - 56) = 344

Add 56 on both sides

b² = 400

b = ±√400

b = ±20

Therefore, the value of b is ±20.

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Find the values of b such that the function has the given maximum value. f(x) = -x² + bx - 14; Maximum value: 86

Summary:

The values of b such that the function has the given maximum value. f(x) = -x² + bx - 14; Maximum value: 86 is ±20.