Find the maximum and minimum values of `f(x)=-|x+1|+3` on `R` , if any, without using derivatives.

Explanation

To find the maximum and minimum values of the function \( f(x) = -|x + 1| + 3 \) on \( \mathbb{R} \) without using derivatives, we can analyze the function step by step.

### Step 1: Understand the Absolute Value Function
The function involves an absolute value, \( |x + 1| \). The absolute value function \( |x + 1| \) is defined as:
- \( x + 1 \) when \( x + 1 \geq 0 \) (i.e., \( x \geq -1 \))
- \( -(x + 1) \) when \( x + 1 < 0 \) (i.e., \( x < -1 \))

### Step 2: Break Down the Function Based on Cases
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