To determine the behavior of $f$ on the interval $0 < x < 3$, we analyze the sign of $f'(x)$, which is given as:

$f'(x) = x^2(x-3)(x-5)^2$

Key Observations:

1. Critical points occur where $f'(x) = 0$.

   • $x^2 = 0 \implies x = 0$ (double root).
   • $x - 3 = 0 \implies x = 3$ (single root).
   • $x - 5 = 0 \implies x = 5$ (double root).

2. Roots and Multiplicities:

   • $x = 0$ and $x = 5$ have even multiplicity (double roots), so the sign of $f'(x)$ does not change at these points.
   • $x = 3$ has odd multiplicity (single root), so the sign of $f'(x)$ does change at this point.

3. Behavior of $f'(x)$ on $0 < x < 3$:

   • For $0 < x < 3$, the term $x^2 > 0$ (since $x^2$ is always non-negative).
   • $x - 3 < 0$ because $x$ is less than 3.
   • $(x-5)^2 > 0$ because it is squared.

   Therefore:
   $f'(x) = x^2 \cdot (x-3) \cdot (x-5)^2 < 0$
   This is because the factor $(x-3)$ is negative in this interval, while the other factors are positive.

Conclusion:

On the interval $0 < x < 3$, $f'(x)$ is negative. This means that the function $f$ is decreasing on this interval.

Correct Answer:

• "it is decreasing"

Would you like a further explanation of critical points or sign charts? Here are 5 related questions to deepen your understanding:

1. How do multiplicities of roots affect the behavior of a derivative function?
2. How do you determine intervals where a function is increasing or decreasing?
3. What does it mean for a function to be concave up or down?
4. How can you use $f'(x)$ to sketch the graph of $f(x)$?
5. What is the difference between critical points and inflection points?

Tip: Always check the sign of $f'(x)$ within intervals defined by critical points to determine where a function increases or decreases.