To find the maximum value of the function \(f\left( x \right) = 1-\left| x \right|\) within the interval \(-1 \le x \le 1\), we need to understand the behavior of the absolute value function, \(|x|\).

Understanding the Absolute Value Function

The absolute value of a number \(x\), denoted as \(|x|\), is its distance from zero on the number line. It is always non-negative:

• If \(x \ge 0\), then \(|x| = x\).
• If \(x < 0\), then \(|x| = -x\).

The minimum value of \(|x|\) is 0, which occurs when \(x=0\).

Analyzing the Function \(f(x) = 1 - |x|\)

The function \(f\left( x \right) = 1-\left| x \right|\) involves subtracting the absolute value of \(x\) from 1. To maximize \(f(x)\), we need to minimize the value of \(|x|\) being subtracted.

Finding the Maximum Value in the Interval \(-1 \le x \le 1\)

We are considering the function within the closed interval \(\left[ -1, 1 \right]\). Let's examine the value of \(|x|\) within this interval:

• The minimum value of \(|x|\) in \(\left[ -1, 1 \right]\) is 0, which occurs at \(x=0\).
• The maximum value of \(|x|\) in \(\left[ -1, 1 \right]\) is 1, which occurs at both \(x=1\) and \(x=-1\).

Since \(f(x) = 1 - |x|\), the function \(f(x)\) will be at its maximum when \(|x|\) is at its minimum.

The minimum value of \(|x|\) in the interval \(-1 \le x \le 1\) is 0, occurring at \(x=0\).

Therefore, the maximum value of the function \(f(x)\) is:

$$ f_{\text{max}} = f(0) = 1 - |0| = 1 - 0 = 1 $$

Evaluating the Function at Interval Endpoints

Let's check the function's value at the endpoints of the interval \(\left[ -1, 1 \right]\):

• At \(x = -1\): $$ f(-1) = 1 - |-1| = 1 - 1 = 0 $$
• At \(x = 1\): $$ f(1) = 1 - |1| = 1 - 1 = 0 $$

Conclusion

Comparing the values:

• \(f(0) = 1\)
• \(f(-1) = 0\)
• \(f(1) = 0\)

The highest value the function attains is 1, which occurs when \(x=0\).

Thus, the value of \(x\) at which the function attains its maximum is 0, and the maximum value is 1.