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.