Consider a function $f\left(x\right)=1-\left|x\right|$ in $-1\le x\le 1$. The value of $x$ at which the function attains a maximum, and the maximum value of the function are:

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.