To find the increasing intervals of the function $f(x) = x^3 - 3x^2 + 4x - 9$, we need to analyze its first derivative.

Step 1: Find the first derivative $f'(x)$

The first derivative of $f(x)$ tells us the slope of the function. If $f'(x) > 0$, the function is increasing. If $f'(x) < 0$, the function is decreasing.

$f'(x) = \frac{d}{dx} \left( x^3 - 3x^2 + 4x - 9 \right)$ Using the power rule of differentiation: $f'(x) = 3x^2 - 6x + 4$

Step 2: Find the critical points

To find the critical points where the function changes from increasing to decreasing (or vice versa), we set the first derivative equal to 0:

$3x^2 - 6x + 4 = 0$

To solve this quadratic equation, use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 3$, $b = -6$, and $c = 4$.

$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(3)(4)}}{2(3)}$ $x = \frac{6 \pm \sqrt{36 - 48}}{6}$ $x = \frac{6 \pm \sqrt{-12}}{6}$

Since the discriminant is negative ($\sqrt{-12}$), there are no real solutions. This means that the first derivative $f'(x) = 3x^2 - 6x + 4$ never equals zero and does not change sign.

Step 3: Analyze the sign of $f'(x)$

Now, we check the sign of $f'(x) = 3x^2 - 6x + 4$ for all values of $x$.

We know that the quadratic $3x^2 - 6x + 4$ is a parabola opening upwards (since the coefficient of $x^2$ is positive). Let's evaluate the value of $f'(x)$ at a test point, say $x = 0$:

$f'(0) = 3(0)^2 - 6(0) + 4 = 4$

Since $f'(0) = 4 > 0$, the first derivative is positive for all values of $x$, meaning that $f(x)$ is always increasing.

Conclusion:

The function $f(x) = x^3 - 3x^2 + 4x - 9$ is increasing for all values of $x$, i.e., the increasing interval is:

$(-\infty, \infty)$

Would you like more details on how this conclusion was reached? Feel free to ask! Here are some related questions to explore:

1. How do you find the decreasing interval of a function using its first derivative?
2. What is the meaning of critical points in the context of derivatives?
3. How can we determine the concavity of the function from the second derivative?
4. What does it mean for a function to have no real critical points?
5. How can the discriminant of a quadratic equation tell us about the nature of its roots?

Tip: Always analyze the sign of the first derivative to understand whether a function is increasing or decreasing. If the first derivative is always positive, the function is increasing everywhere!