To find the intervals where $f(x) > 0$ for the function $f(x) = x^4 - 2x^3 - 3x^2$, we need to determine the regions where the function is positive. Here are the steps:

1. Factor the function: First, factor $f(x)$ completely.

$f(x) = x^4 - 2x^3 - 3x^2 = x^2(x^2 - 2x - 3)$

Now, factor the quadratic expression:

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

So, the fully factored form is:

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

2. Find the critical points: Set $f(x) = 0$ to find the critical points.

$x^2 (x - 3) (x + 1) = 0$

This gives:

$x = 0, \quad x = 3, \quad x = -1$

3. Determine the sign of $f(x)$ in each interval: We will test the sign of $f(x)$ in the intervals determined by the critical points: $(-\infty, -1)$, $(-1, 0)$, $(0, 3)$, and $(3, \infty)$.

• For $x \in (-\infty, -1)$: Choose $x = -2$.

  $f(-2) = (-2)^2 \times (-2 - 3) \times (-2 + 1) = 4 \times (-5) \times (-1) = 20 \quad (\text{positive})$

• For $x \in (-1, 0)$: Choose $x = -0.5$.

  $f(-0.5) = (-0.5)^2 \times (-0.5 - 3) \times (-0.5 + 1) = 0.25 \times (-3.5) \times 0.5 = -0.4375 \quad (\text{negative})$

• For $x \in (0, 3)$: Choose $x = 1$.

  $f(1) = 1^2 \times (1 - 3) \times (1 + 1) = 1 \times (-2) \times 2 = -4 \quad (\text{negative})$

• For $x \in (3, \infty)$: Choose $x = 4$.

  $f(4) = 4^2 \times (4 - 3) \times (4 + 1) = 16 \times 1 \times 5 = 80 \quad (\text{positive})$

4. Identify the intervals where $f(x) > 0$:

From the analysis:

• $f(x) > 0$ in the intervals $(-\infty, -1)$ and $(3, \infty)$.

Therefore, the intervals where $f(x) > 0$ are:

$(-\infty, -1) \cup (3, \infty)$

Would you like more details or have any questions?

Here are some related questions:

1. How do we find the intervals where $f(x) < 0$?
2. What are the turning points of $f(x) = x^4 - 2x^3 - 3x^2$?
3. How can we sketch the graph of $f(x) = x^4 - 2x^3 - 3x^2$?
4. What is the behavior of $f(x)$ as $x$ approaches infinity?
5. How do we determine the local minima and maxima of $f(x)$?

Tip: To check where a polynomial is positive or negative, factor it completely and test the sign of each factor in the intervals determined by the roots!