To sketch the curve $y = x^3 + 3x^2$, let’s analyze the function step by step.

1. Rewrite the Function

The function can be rewritten as: $y = x^2(x + 3)$ This form helps us identify the roots and factor the expression.

2. Find the Roots (x-intercepts)

Set $y = 0$: $x^2(x + 3) = 0$ This gives us roots at: $x = 0 \quad \text{(with multiplicity 2)} \quad \text{and} \quad x = -3$ These are the points where the curve intersects the x-axis.

3. Determine the First Derivative (y')

To find the critical points and determine where the function is increasing or decreasing, calculate $y'$: $y' = \frac{d}{dx}(x^3 + 3x^2) = 3x^2 + 6x = 3x(x + 2)$ Setting $y' = 0$ gives: $3x(x + 2) = 0$ This results in critical points at: $x = 0 \quad \text{and} \quad x = -2$

4. Analyze the Critical Points and Intervals

To determine the behavior of the function around the critical points, we’ll analyze the sign of $y'$ in each interval:

• For $x < -2$: $y' > 0$ (the function is increasing).
• For $-2 < x < 0$: $y' < 0$ (the function is decreasing).
• For $x > 0$: $y' > 0$ (the function is increasing again).

This tells us that:

• $x = -2$ is a local maximum.
• $x = 0$ is a local minimum.

5. Determine the Second Derivative (y'')

To analyze concavity, calculate $y''$: $y'' = \frac{d}{dx}(3x^2 + 6x) = 6x + 6$ Setting $y'' = 0$ gives: $6x + 6 = 0 \Rightarrow x = -1$

• For $x < -1$: $y'' < 0$ (concave down).
• For $x > -1$: $y'' > 0$ (concave up).

This means there is an inflection point at $x = -1$.

6. Sketch the Curve

Based on this analysis:

• The curve crosses the x-axis at $x = 0$ and $x = -3$.
• It has a local maximum at $x = -2$ and a local minimum at $x = 0$.
• There is an inflection point at $x = -1$.

This information allows us to sketch the general shape of the curve.

Would you like a detailed plot or have further questions on the steps?

Here are five related questions for deeper understanding:

1. How do you determine whether a critical point is a maximum, minimum, or neither?
2. What is the significance of an inflection point in a curve?
3. Why is it helpful to analyze the first and second derivatives when sketching a curve?
4. How can you confirm if a root has a multiplicity and its impact on the curve?
5. What role does concavity play in understanding the behavior of a function?

Tip: When analyzing curves, always look for symmetry and check if the function is even, odd, or neither; this can simplify the sketching process.