Find the stationary points and rough points and endpoint for f(x)=x^2+3-1$$ \le x\le4 $$

Let's solve this step-by-step. We are asked to find the **stationary points, rough points (discontinuities or non-differentiable points), and endpoints** of the function $ f(x) = x^2 + 3 $ on the closed interval $ -1 \leq x \leq 4 $.

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### Step 1: Stationary Points
Stationary points occur when the **first derivative** of the function is zero.

1. Compute the derivative of $ f(x) $:
   \[
   f'(x) = \frac{d}{dx}(x^2 + 3) = 2x
   \]
2. Set the derivative equal to zero to find the critical points:
   \[
   2x = 0 \quad \Rightarrow \quad x = 0
   \]

Thus, the only stationary point within the interval is $ x = 0 $.

3. Evaluate the function at $ x = 0 $:
   \[
   f(0) = 0^2 + 3 = 3
   \]

So, the stationary point is $ (0, 3) $.

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### Step 2: Endpoints
Since the function is defined on a **closed interval** $ -1 \leq x \leq 4 $, we must evaluate it at the endpoints.

- At $ x = -1 $:
  \[
  f(-1) = (-1)^2 + 3 = 4
  \]

- At $ x = 4 $:
  \[
  f(4) = 4^2 + 3 = 16 + 3 = 19
  \]

Thus, the function takes the values:
- At $ x = -1 $: $ f(-1) = 4 $
- At $ x = 4 $: $ f(4) = 19 $

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### Step 3: Non-differentiable Points (Rough Points)
- The function $ f(x) = x^2 + 3 $ is a **polynomial**, which is continuous and differentiable everywhere on $ \mathbb{R} $. 
- Thus, there are **no rough points** or non-differentiable points in the given interval $ -1 \leq x \leq 4 $.

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### Summary
- **Stationary point**: $ (0, 3) $
- **Endpoints**: $ (-1, 4) $ and $ (4, 19) $
- **No rough points** (the function is differentiable throughout the interval).

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### Optional: Graphing Insight
The function $ f(x) = x^2 + 3 $ is a **parabola that opens upwards**. 
- The stationary point $ (0, 3) $ is the **global minimum** in the given interval.
- The values increase as we move towards the endpoints.

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Do you want further clarifications or have any questions?

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Here are five related questions to deepen understanding:
1. What is the difference between a stationary point and an inflection point?
2. How do we determine whether a stationary point is a minimum or maximum?
3. What happens if the derivative is undefined at some point in the interval?
4. How do we find global extrema for functions defined on closed intervals?
5. How do endpoints affect the extrema in a given interval?

**Tip**: Always check both endpoints and stationary points when finding extrema on a closed interval.

Find the stationary points and rough points and endpoint for f(x)=x^2+3, -1 ≤ x ≤ 4

Learn how to find stationary points, endpoints, and assess differentiability for the function f(x) = x^2 + 3 on the interval [-1, 4]. This solution includes key concepts from calculus, such as finding the derivative, solving for stationary points, and checking for rough points. Suitable for high school students in Grades 10-12.

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