Find the maximum or minimum values, if any, of the following funcitons without using the derivatives :
`f(x)=x+1,.x in [-1, 1]`

Answer

To find the maximum and minimum values of the function \( f(x) = x + 1 \) on the interval \( x \in [-1, 1] \) without using derivatives, we can follow these steps:

### Step 1: Identify the function and the interval
The function we are working with is:
\[ f(x) = x + 1 \]
and the interval is \( x \in [-1, 1] \).

### Step 2: Evaluate the function at the endpoints of the interval
We will calculate the value of the function at the endpoints of the interval, which are \( x = -1 \) and \( x = 1 \).

1. For \( x = -1 \):
\[ f(-1) = -1 + 1 = 0 \]

2. For \( x = 1 \):
\[ f(1) = 1 + 1 = 2 \]

### Step 3: Identify the values at critical points within the interval
Since \( f(x) = x + 1 \) is a linear function, it does not have any critical points within the interval. Therefore, we only need to consider the values we calculated at the endpoints.

### Step 4: Compare the values
Now we compare the values obtained:
- \( f(-1) = 0 \)
- \( f(1) = 2 \)

### Step 5: Determine the maximum and minimum values
From the values calculated:
- The minimum value of \( f(x) \) on the interval \( [-1, 1] \) is \( 0 \) (at \( x = -1 \)).
- The maximum value of \( f(x) \) on the interval \( [-1, 1] \) is \( 2 \) (at \( x = 1 \)).

### Final Answer
- Minimum value: \( 0 \)
- Maximum value: \( 2 \)

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