Question

consider the function \\(f(x) = 2 - 2x^2\\), \\(-5 \le x \le 2\\).

the absolute maximum value is
and this occurs at \\(x =\\)

the absolute minimum value is
and this occurs at \\(x =\\)

Explanation:

Find the critical points of the function

Using the Critical Points knowledge point

$$
LATEXBLOCK0
$$

Since \(0 \in [-5, 2]\), \(x = 0\) is a valid critical point.

Evaluate the function at critical points and endpoints

Using the Absolute Extrema on Closed Intervals knowledge point

$$
LATEXBLOCK1
$$

Identify the absolute maximum and minimum values

Using the Absolute Extrema on Closed Intervals knowledge point

$$
LATEXBLOCK2
$$

Answer:

Consider the function \(f(x) = 2 - 2x^2\), \(-5 \le x \le 2\).

The absolute maximum value is <blank>2</blank>
and this occurs at \(x =\) <blank>0</blank>.

The absolute minimum value is <blank>-48</blank>
and this occurs at \(x =\) <blank>-5</blank>.