Question

Rogelio is asked to sketch a graph of g(x)=3x^3(x-5)^2(5-x)^4. How many turning points should his sketch include?

Answer 1

To determine how many turning points the function \( g(x) = 3x^3(x-5)^2(5-x)^4 \) has, we need to analyze its derivatives and critical points.

The function is a polynomial, so we can find its turning points by looking at its first derivative \( g'(x) \).

1. **Finding the degree of the polynomial:**
- The polynomial \( g(x) \) can be expressed as:
\[
g(x) = 3x^3 \cdot (x-5)^2 \cdot (5-x)^4
\]
- The degree of \( g(x) \) is the sum of the degrees of its factors:
- The degree of \( 3x^3 \) is 3.
- The degree of \( (x-5)^2 \) is 2.
- The degree of \( (5-x)^4 \) is 4.
- Therefore, the total degree of \( g(x) \) is \( 3 + 2 + 4 = 9 \).

2. **Finding the number of turning points:**
- The maximum number of turning points of a polynomial of degree \( n \) is \( n - 1 \).
- In this case, since the degree of \( g(x) \) is 9, the maximum number of turning points is \( 9 - 1 = 8 \).

3. **Counting the actual turning points:**
- However, we also need to consider the multiplicity of the roots:
- The root \( x = 0 \) has a multiplicity of 3.
- The root \( x = 5 \) has a multiplicity of 2.
- The root \( x = 5 \) (from the factor \( (5-x)^4 \)) is also related, as it will affect the symmetry and where the derivatives change signs.
- The behavior of the function near these roots and their multiplicity affects whether they contribute to turning points.

Since \( x = 0 \) (multiplicity 3, not a turning point), \( x = 5 \) (multiplicity 2, contributes turning point), and \( x = 5 \) from \( (5-x)^4 \) (this root contributes a local maximum behavior).

Thus, \( g(x) \) will have \( 8 \) turning points in total but checking for multiplicity where the roots are located effectively will narrow down actual contributing turning points.

### Conclusion:
The maximum possible number of turning points for the function \( g(x) = 3x^3(x-5)^2(5-x)^4 \) is **8**, although there may be fewer actual turning points depending on the nature of the roots and multiplicity.