Yes, let’s analyze and solve exercise 3 based on the information provided in the uploaded image.

Problem Analysis:

For the function $f(x) = x e^{x^2}$, we are tasked to:

1. Determine the intervals where the function is concave or convex.
2. Identify the inflection points (buigpunten).

Solution Steps:

Step 1: Find the first derivative $f'(x)$

Using the product rule and chain rule for differentiation, we get: $f(x) = x \cdot e^{x^2}$ $f'(x) = e^{x^2} \cdot (2x^2 + 1)$

Step 2: Find the second derivative $f''(x)$

To determine concavity, we need the second derivative: $f''(x) = 2x \cdot e^{x^2} \cdot (2x^2 + 3)$

Step 3: Analyze concavity

1. The function is concave (downwards) when $f''(x) < 0$. This occurs when $x < 0$.
2. The function is convex (upwards) when $f''(x) > 0$. This occurs when $x > 0$.

Step 4: Identify inflection points

An inflection point occurs where the concavity changes, which means at $x = 0$ (since the function changes from concave to convex here).

Thus, the function has an inflection point at $(0, 0)$.

Summary:

• The function $f(x) = x e^{x^2}$ is concave for $x < 0$ and convex for $x > 0$.
• There is an inflection point at $(0, 0)$.

Would you like further details on any step, or do you have specific questions about the solution?

Here are five related questions you might consider:

1. How can the product rule be applied in other contexts?
2. What are common methods to verify inflection points?
3. How does convexity relate to real-world applications?
4. Can higher derivatives provide insights into the behavior of functions?
5. How would the solution change if the function had additional terms?

Tip: Understanding concavity and convexity can help in sketching graphs and analyzing the behavior of functions over different intervals.