Question

by inspecting the graph of the function, find the absolute maximum and absolute minimum on the given interval.
g(x)=x² - 2 on -5,4
choose the correct graph below.
find the absolute minimum. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute minimum is at x=
(use a comma to separate answers as needed.)
b. there is no absolute minimum.

Explanation:

Step1: Analyze the function \(g(x)=x^{2}-2\)

The function \(g(x)=x^{2}-2\) is a parabola. The general form of a parabola is \(y = ax^{2}+bx + c\), here \(a = 1\), \(b=0\), \(c=-2\). Since \(a=1>0\), the parabola opens upwards. The vertex of the parabola \(y = ax^{2}+bx + c\) is at \(x=-\frac{b}{2a}\). Substituting \(a = 1\) and \(b = 0\), we get \(x = 0\). When \(x = 0\), \(g(0)=0^{2}-2=-2\).

Step2: Evaluate the function at the endpoints of the interval \([-5,4]\)

When \(x=-5\), \(g(-5)=(-5)^{2}-2=25 - 2=23\).
When \(x = 4\), \(g(4)=4^{2}-2=16-2 = 14\).

Answer:

The absolute minimum is \(-2\) at \(x = 0\).