Determine the value of x at which -x ^ 2 + 8-1 is greatest.

Method I.

y = -x² + 8x – 1. Consider a function, it is a quadratic parabola, its branches are located downward.

The highest point of the graph will be the apex of the parabola. The x value at this point will be the maximum. Find the x coordinates of the vertex of the parabola:

x0 = (-b) / 2a = -8 / (- 2) = 4.

This means that at the point x = 4 the value of the function will be maximum. Find the value of the function at this point (that is, the value of y).

y = -4² + 8 * 4 – 1 = -16 + 32 – 1 = 15.

The maximum value of the function is 15.

Method II.

Find the derivative of the function:

y = -x² + 8x – 1.

y ‘= -2x + 8.

Find the zeros of the derivative:

-2x + 8 = 0.

-2x = -8.

x = -8: (-2).

x = 4.

Let us determine the signs of the derivative on each interval:

(-∞; 4) let x = 0; y ‘(0) = -2 * 0 + 8 = 0 + 8 = 8 (plus), the function is increasing.

(4; + ∞) let x = 5; y ‘(5) = -2 * 5 + 8 = -10 + 8 = -2 (minus), the function decreases.

Hence, x = 4 is the maximum point.

Let’s find the maximum value of the function at this point:

y (4) = -4² + 8 * 4 – 1 = -16 + 32 – 1 = 15.

Answer: The largest value of the function is 15.