Question

What is the absolute minimum value of g(x) = -²x² - ²x - ¹ over [-³, ⁴]?

a) -⁹
b) -¹²
c) -¹⁷
d) -⁵

Answer 1

The absolute minimum value of g(x) = -²x² - ²x - ¹ over the interval

`[3,4][3 , 4 ]` is c) -¹⁷.

Step-by-step explanation:

To find the absolute minimum of the quadratic function g(x) = -²x² - ²x - ¹, we first locate its critical points by finding where the derivative is zero. Taking the derivative and setting it equal to zero gives us g'(x) = -4x - 2 = 0. Solving for x, we find x = -½, which is the only critical point.

To determine whether this critical point corresponds to a minimum, maximum, or neither, we examine the sign of the second derivative. Calculating g''(x), we get g''(x) = -4. Since the second derivative is negative, the critical point x = -½ corresponds to a local maximum.

To find the absolute minimum over the given interval

`[-3,4][ 3 , 4`]

], we evaluate g(x) at the endpoints and the critical point. g(-3) = -14, g(-½) = -17/4, and g(4) = -33. The smallest of these values is -17/4, which corresponds to option c) -¹⁷. Therefore, -¹⁷ is the absolute minimum of g(x) over the interval

`[-3,4][-3 , 4 ].`