) Show that the function f: [-1,2]-R defined by f(x) = x ^ 2 + 2x + 4 is an increasing function.

To show that the function \( f(x) = x^2 + 2x + 4 \), defined on the interval \( [-1, 2] \), is increasing, we need to analyze its derivative. A function is increasing if its derivative is non-negative (i.e., \( f'(x) \geq 0 \)) for all \( x \) in the given interval.

### Step 1: Find the derivative of \( f(x) \)
The derivative of \( f(x) = x^2 + 2x + 4 \) with respect to \( x \) is:

\[
f'(x) = \frac{d}{dx}(x^2 + 2x + 4) = 2x + 2
\]

### Step 2: Analyze the sign of \( f'(x) \)
The derivative \( f'(x) = 2x + 2 \) is a linear function. To check whether it is non-negative on the interval \( [-1, 2] \), let's evaluate its behavior at the endpoints and within the interval:

- At \( x = -1 \):
  \[
  f'(-1) = 2(-1) + 2 = 0
  \]

- At \( x = 2 \):
  \[
  f'(2) = 2(2) + 2 = 6
  \]

Thus, the derivative is 0 at \( x = -1 \) and 6 at \( x = 2 \).

### Step 3: General behavior of \( f'(x) \)
Since \( f'(x) = 2x + 2 \) is a linear function with a positive slope (2), it increases as \( x \) increases. More importantly, for any \( x \) in the interval \( [-1, 2] \), the derivative \( f'(x) \geq 0 \).

In particular:
- For \( x = -1 \), \( f'(x) = 0 \), which indicates a horizontal tangent (the function is neither increasing nor decreasing at that point).
- For \( x > -1 \), \( f'(x) > 0 \), indicating that the function is strictly increasing.

### Conclusion:
Since \( f'(x) \geq 0 \) for all \( x \in [-1, 2] \), we can conclude that \( f(x) \) is an increasing function on the interval \( [-1, 2] \).

Would you like more details on how the derivative affects function behavior?

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**Related Questions:**
1. What does it mean for a function to be strictly increasing versus increasing?
2. Can a function be increasing if its derivative is 0 at some point?
3. How does concavity affect the increasing or decreasing nature of a function?
4. How can the second derivative provide additional information about function behavior?
5. What is the relationship between the derivative and the graph of a function?

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

Show that the function f: [-1,2] -> R defined by f(x) = x^2 + 2x + 4 is an increasing function.

Explore how to prove that the quadratic function f(x) = x^2 + 2x + 4 is increasing on the interval [-1, 2] using its derivative. This solution involves calculus concepts and applies the increasing function theorem, suitable for high school students in Grades 10-12.

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