Find the maximum or minimum values, if any, of the following funcitons without using the derivatives :
`g(x)=-|x-1|+3.`

Answer

To find the maximum or minimum values of the function \( g(x) = -|x-1| + 3 \) without using derivatives, we can analyze the function step by step.

### Step 1: Understand the Absolute Value Function
The function \( |x-1| \) represents the distance of \( x \) from 1. It is always non-negative, meaning \( |x-1| \geq 0 \) for all \( x \).

### Step 2: Determine the Minimum Value of \( |x-1| \)
The minimum value of \( |x-1| \) occurs when \( x = 1 \). At this point, \( |x-1| = 0 \).

### Step 3: Substitute the Minimum Value into \( g(x) \)
Now, substitute \( x = 1 \) into the function \( g(x) \):
\[ g(1) = -|1-1| + 3 = -0 + 3 = 3 \]

### Step 4: Analyze the Behavior of \( g(x) \)
As \( x \) moves away from 1 (either to the left or right), the value of \( |x-1| \) increases. Since \( g(x) \) has a negative sign in front of the absolute value, \( -|x-1| \) will decrease as \( |x-1| \) increases. Therefore, \( g(x) \) will decrease from the maximum value of 3.

### Step 5: Conclusion on Maximum and Minimum Values
- The maximum value of \( g(x) \) is \( 3 \), which occurs at \( x = 1 \).
- There is no minimum value for \( g(x) \) since as \( |x-1| \) increases indefinitely, \( g(x) \) will continue to decrease without bound.

### Final Answer:
- Maximum value: \( 3 \) at \( x = 1 \)
- Minimum value: None (it decreases indefinitely)