Show that the function f: [1,4]→R defined by f(x) = x ^ 3 - 3x is an increasing function

To show that the function \( f(x) = x^3 - 3x \) is an increasing function on the interval \( [1, 4] \), we need to examine its derivative, as the derivative of a function tells us about its rate of change.

### Step 1: Find the derivative of \( f(x) \)
The derivative of \( f(x) = x^3 - 3x \) is:

\[
f'(x) = \frac{d}{dx}(x^3 - 3x) = 3x^2 - 3
\]

### Step 2: Analyze the sign of \( f'(x) \)
Now, we need to determine whether \( f'(x) \geq 0 \) for all \( x \in [1, 4] \). This will tell us if \( f(x) \) is increasing throughout the interval.

\[
f'(x) = 3(x^2 - 1) = 3(x - 1)(x + 1)
\]

Now, let's analyze the sign of \( f'(x) = 3(x - 1)(x + 1) \) on the interval \( [1, 4] \).

- At \( x = 1 \), \( f'(1) = 3(1 - 1)(1 + 1) = 0 \).
- For \( x > 1 \), since \( x - 1 > 0 \) and \( x + 1 > 0 \), \( f'(x) > 0 \).

Thus, for \( x \in [1, 4] \), \( f'(x) \geq 0 \), meaning the function is non-decreasing, and since \( f'(x) > 0 \) for \( x > 1 \), it is strictly increasing on \( (1, 4] \).

### Conclusion:
The function \( f(x) = x^3 - 3x \) is increasing on the interval \( [1, 4] \) because its derivative is non-negative and positive for \( x > 1 \).

Would you like more details or have any questions?

Here are 5 related questions you might want to explore:
1. What is the difference between a strictly increasing and a non-decreasing function?
2. How does the second derivative of a function help in determining concavity?
3. Can a function be increasing even if its derivative is zero at some points?
4. How do we classify critical points in a function like \( f(x) \)?
5. What is the importance of the Mean Value Theorem in understanding function behavior?

**Tip:** Always check the derivative of a function over a specific interval to determine whether the function is increasing, decreasing, or constant.

Discover how to prove that the function f(x) = x^3 - 3x is increasing on the interval [1,4] using calculus. This solution involves finding the derivative of the function and applying the First Derivative Test. Ideal for students in Grades 11-12.

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