Finding Minimum and Maximum Values of \( f(x) = |2 - |x|| \)

We need to find the minimum and maximum values of the function \( f(x) = |2 - |x|| \) over the closed interval \( -3 \leq x \leq 3 \). The domain for \( x \) is the interval \( [-3, 3] \).

Analyzing the Function \( f(x) = |2 - |x|| \)

The function involves nested absolute values. Let's break it down by considering the range of \( |x| \) for \( x \in [-3, 3] \).

For \( x \in [-3, 3] \), the value of \( |x| \) ranges from a minimum of 0 (at \( x=0 \)) to a maximum of 3 (at \( x=3 \) or \( x=-3 \)). So, \( 0 \leq |x| \leq 3 \).

Next, consider the expression inside the outer absolute value: \( 2 - |x| \). Since \( 0 \leq |x| \leq 3 \), we can find the range of \( 2 - |x| \):

• When \( |x| = 0 \), \( 2 - |x| = 2 - 0 = 2 \).
• When \( |x| = 3 \), \( 2 - |x| = 2 - 3 = -1 \).

As \( |x| \) increases from 0 to 3, \( 2 - |x| \) decreases from 2 to -1. Therefore, the range of \( 2 - |x| \) for \( x \in [-3, 3] \) is the interval \( [-1, 2] \).

Now, we need to find the range of \( f(x) = |2 - |x|| \), which is the absolute value of the expression \( 2 - |x| \). The values of \( 2 - |x| \) lie in the interval \( [-1, 2] \). We are looking for the minimum and maximum values of \( |y| \) where \( y \in [-1, 2] \).

• The minimum value of \( |y| \) for \( y \in [-1, 2] \) is 0, which occurs when \( y = 0 \).
• The maximum value of \( |y| \) for \( y \in [-1, 2] \) occurs at the endpoints of the interval or where the expression is 0. The values of \( |y| \) for \( y \in \{-1, 2\} \) are \( |-1| = 1 \) and \( |2| = 2 \). The maximum value is 2.

So, the range of \( f(x) = |2 - |x|| \) for \( x \in [-3, 3] \) is \( [0, 2] \).

Determining Minimum and Maximum Values

Based on the analysis, the minimum value of \( f(x) \) is 0 and the maximum value is 2.

• The minimum value \( m = 0 \) occurs when \( 2 - |x| = 0 \), which means \( |x| = 2 \). This happens at \( x = 2 \) and \( x = -2 \), both of which are in the interval \( [-3, 3] \).
• The maximum value \( M = 2 \) occurs when \( 2 - |x| = 2 \) or \( 2 - |x| = -2 \).
  • \( 2 - |x| = 2 \implies |x| = 0 \implies x = 0 \). At \( x=0 \), \( f(0) = |2 - |0|| = |2| = 2 \).
  • \( 2 - |x| = -2 \implies |x| = 4 \). This has no solution for \( x \) in the interval \( [-3, 3] \) because the maximum value of \( |x| \) is 3.

Let's also check the values at the endpoints of the domain \( x = \pm 3 \):

• \( f(3) = |2 - |3|| = |2 - 3| = |-1| = 1 \)
• \( f(-3) = |2 - |-3|| = |2 - 3| = |-1| = 1 \)

Comparing the values found: \( f(0)=2 \), \( f(\pm 2)=0 \), \( f(\pm 3)=1 \). The values are 0, 1, and 2.

The smallest value is 0, so \( m = 0 \).

The largest value is 2, so \( M = 2 \).

Thus, the minimum value is \( m = 0 \) and the maximum value is \( M = 2 \).

The table shows the function values at critical points and endpoints within the interval \( [-3, 3] \).

The minimum value observed is 0, and the maximum value observed is 2.

Conclusion on Minimum and Maximum Values

The minimum value is \( m = 0 \) and the maximum value is \( M = 2 \).

Revision Table: Minimum and Maximum Values

Additional Information: Finding Extrema of Functions

To find the minimum and maximum values (extrema) of a continuous function \( f(x) \) on a closed interval \( [a, b] \), you should follow these steps:

1. Find the critical points of \( f(x) \) within the interval \( (a, b) \). Critical points are where \( f'(x) = 0 \) or \( f'(x) \) is undefined.
2. Evaluate the function \( f(x) \) at each critical point found in step 1 that lies within the interval \( (a, b) \).
3. Evaluate the function \( f(x) \) at the endpoints of the interval, i.e., find \( f(a) \) and \( f(b) \).
4. The minimum value of \( f(x) \) on \( [a, b] \) is the smallest of the values calculated in steps 2 and 3.
5. The maximum value of \( f(x) \) on \( [a, b] \) is the largest of the values calculated in steps 2 and 3.

For functions with absolute values, finding the derivative might be tricky. An alternative approach, especially for functions like \( |g(x)| \), is to analyze the range of the inner function \( g(x) \) over the given interval and then find the minimum and maximum of the absolute value of that range. In this problem, the inner function is \( 2 - |x| \).