x^2+2x+1 The function is increasing on the interval 

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!

x^2+2x+1 The function is increasing on the interval

Find the interval on which the function f(x) = x^2 + 2x + 1 is increasing by using the first derivative test. This solution walks through finding the derivative, analyzing it, and solving inequalities, suitable for students in Grades 9-12.

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