To show that the function $f(x) = x^3 - 3x$ is increasing on the interval $[1,4]$, we need to examine the behavior of its derivative. A function is increasing on an interval if its derivative is non-negative throughout that interval.

Step 1: Compute 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 $f'(x)$

We need to determine where $f'(x) = 3x^2 - 3$ is positive, negative, or zero.

First, let's factor the derivative:

$f'(x) = 3(x^2 - 1) = 3(x - 1)(x + 1)$

Now, let's check the critical points by setting $f'(x) = 0$:

$3(x - 1)(x + 1) = 0$

This equation gives two critical points:

$x = 1 \quad \text{and} \quad x = -1$

However, since the function is defined on the interval $[1,4]$, we are only concerned with $x = 1$ in this interval.

Step 3: Determine the sign of $f'(x)$ on the interval $[1,4]$

Now, let's examine the sign of $f'(x) = 3(x - 1)(x + 1)$ on the interval $[1, 4]$:

• At $x = 1$, $f'(x) = 3(1 - 1)(1 + 1) = 0$.
• For $x > 1$ (in particular, for $x \in (1, 4]$), both $(x - 1) > 0$ and $(x + 1) > 0$, so $f'(x) > 0$.

Thus, $f'(x) \geq 0$ for all $x \in [1, 4]$, with equality only at $x = 1$.

Conclusion:

Since $f'(x) \geq 0$ on $[1,4]$, the function $f(x) = x^3 - 3x$ is increasing on the interval $[1,4]$.

Would you like further clarification on any step or more details?

Here are some related questions for further exploration:

1. How do critical points influence the behavior of a function?
2. What does it mean for a function to be strictly increasing versus non-decreasing?
3. How can the second derivative provide more insights into the nature of a function's increase or decrease?
4. What is the role of endpoints when determining if a function is increasing or decreasing on a closed interval?
5. Can a function be increasing on an interval but have a flat (zero derivative) point within that interval?

Tip: A derivative being zero at a point does not mean the function decreases or increases—it just indicates a potential critical point (like a local maximum, minimum, or inflection point). Always check the sign of the derivative around critical points!