To determine the intervals where the function $f(x) = 3x^4 - 4x^3 - 12x^2 + 5$ is increasing or decreasing, we need to follow these steps:

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

The first derivative $f'(x)$ will give us information about the slope of the function. We can use it to determine where the function is increasing (positive slope) and where it is decreasing (negative slope).

$f'(x) = \frac{d}{dx}(3x^4 - 4x^3 - 12x^2 + 5)$

Using the power rule:

$f'(x) = 12x^3 - 12x^2 - 24x$

Step 2: Find the critical points

To find the critical points, set $f'(x) = 0$ and solve for $x$:

$12x^3 - 12x^2 - 24x = 0$

Factor the equation:

$12x(x^2 - x - 2) = 0$

Now, factor the quadratic expression $x^2 - x - 2$:

$12x(x - 2)(x + 1) = 0$

So, the critical points are:

$x = 0, \, x = 2, \, x = -1$

Step 3: Determine the sign of $f'(x)$ in the intervals

Now, we use these critical points to divide the number line into intervals. We will test the sign of $f'(x)$ in each interval to determine where the function is increasing and where it is decreasing.

The intervals are:

• $(-\infty, -1)$
• $(-1, 0)$
• $(0, 2)$
• $(2, \infty)$

Choose test points within each interval, like $x = -2$, $x = -0.5$, $x = 1$, and $x = 3$, and plug them into $f'(x) = 12x(x - 2)(x + 1)$.

• For $x = -2$, $f'(-2) = 12(-2)((-2)-2)((-2)+1) = 12(-2)(-4)(-1) = -96$ (negative)
• For $x = -0.5$, $f'(-0.5) = 12(-0.5)((-0.5)-2)((-0.5)+1) = 12(-0.5)(-2.5)(0.5) = 7.5$ (positive)
• For $x = 1$, $f'(1) = 12(1)((1)-2)((1)+1) = 12(1)(-1)(2) = -24$ (negative)
• For $x = 3$, $f'(3) = 12(3)((3)-2)((3)+1) = 12(3)(1)(4) = 144$ (positive)

Step 4: Determine the intervals of increase and decrease

From the test points, we see:

• $f(x)$ is decreasing on $(-\infty, -1)$ and $(0, 2)$ (where $f'(x) < 0$).
• $f(x)$ is increasing on $(-1, 0)$ and $(2, \infty)$ (where $f'(x) > 0$).

Final Answer:

• The function is increasing on the intervals $(-1, 0)$ and $(2, \infty)$.
• The function is decreasing on the intervals $(-\infty, -1)$ and $(0, 2)$.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you find the concavity of a function using the second derivative?
2. What is the significance of critical points in a function’s graph?
3. How can you classify critical points as local maxima or minima?
4. What happens when a derivative test fails (e.g., a derivative does not exist)?
5. How does the degree of a polynomial affect the shape of its graph?

Tip: For polynomials, always check the behavior as $x \to \infty$ and $x \to -\infty$ to better understand the function’s long-term behavior.