To determine the interval on which the function $f(x) = x^2 + 2x + 1$ is increasing, we first find its derivative and analyze its behavior.

Step 1: Find the derivative

The derivative of $f(x) = x^2 + 2x + 1$ is: $f'(x) = 2x + 2$

Step 2: Analyze the derivative

The function increases where the derivative is positive, that is, $f'(x) > 0$. Let's solve the inequality: $2x + 2 > 0$ $2x > -2$ $x > -1$

Conclusion:

The function $f(x) = x^2 + 2x + 1$ is increasing on the interval $(-1, \infty)$.

Would you like a detailed explanation of how to find intervals where a function is increasing or decreasing?

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Here are five related questions to deepen understanding:

1. How do you find the decreasing intervals of a quadratic function?
2. What is the minimum value of the function $f(x) = x^2 + 2x + 1$?
3. Can you explain the role of the vertex in determining intervals of increase and decrease?
4. How would you apply this method to a cubic function?
5. How does the second derivative relate to concavity and intervals of increase or decrease?

Tip: Always check the sign of the first derivative to determine where a function is increasing or decreasing!