Find the maximum or minimum values, if any, of the following funcitons without using the derivatives :
`f(x)=|x+2|-1`

Answer

To find the maximum or minimum values of the function \( f(x) = |x + 2| - 1 \) without using derivatives, we can analyze the function step by step.

### Step 1: Understand the Absolute Value Function
The function \( f(x) \) consists of an absolute value term \( |x + 2| \). The absolute value function \( |x + 2| \) represents the distance of \( x + 2 \) from 0 on the number line. This distance is always non-negative, meaning \( |x + 2| \geq 0 \) for all \( x \).

### Step 2: Determine the Minimum Value of \( |x + 2| \)
The minimum value of \( |x + 2| \) occurs when \( x + 2 = 0 \). Solving for \( x \):
\[ x + 2 = 0 \implies x = -2 \]
At this point, the value of the absolute function is:
\[ |x + 2| = |0| = 0 \]

### Step 3: Substitute Back into the Function
Now, we substitute \( x = -2 \) back into the function \( f(x) \):
\[ f(-2) = |(-2) + 2| - 1 = |0| - 1 = 0 - 1 = -1 \]

### Step 4: Analyze the Behavior of the Function
Since \( |x + 2| \) is always non-negative, the function \( f(x) = |x + 2| - 1 \) will be:
\[ f(x) \geq 0 - 1 = -1 \]
This means the function \( f(x) \) can never go below \(-1\).

### Step 5: Conclusion
Thus, the minimum value of \( f(x) \) is \(-1\), which occurs at \( x = -2 \). There is no maximum value for \( f(x) \) since as \( |x + 2| \) increases (as \( x \) moves away from \(-2\)), \( f(x) \) will also increase indefinitely.

### Final Answer
- Minimum value of \( f(x) \) is \(-1\) at \( x = -2 \).
- There is no maximum value.