Let \[h\] be a polynomial function and let \[h'\], its **derivative**, be defined as \[h'(x)=x^2(x-2)^2(x-1)^2 \].

**At how many points does the graph of \[h\] have a relative *maximum* ?**

To determine the number of points where the graph of the polynomial function \( h(x) \) has a relative maximum, we need to analyze the behavior of its derivative \( h'(x) \), specifically where it changes sign from positive to negative. 

Given:
\[
h'(x) = x^2(x-2)^2(x-1)^2
\]

### Step 1: Identify the Critical Points
Critical points occur where the derivative is zero or undefined. Since \( h'(x) \) is a polynomial, it's zero at the roots:
- \( x = 0 \)
- \( x = 1 \)
- \( x = 2 \)

### Step 2: Analyze the Sign of \( h'(x) \) Around the Critical Points
We check the sign of \( h'(x) \) around each critical point to determine where the function \( h(x) \) has relative maxima.

- **Factor \( x^2 \):** This factor is always non-negative and equals zero at \( x = 0 \). It doesn't change sign around \( x = 0 \).
- **Factor \( (x-1)^2 \):** This factor is always non-negative and equals zero at \( x = 1 \). It doesn't change sign around \( x = 1 \).
- **Factor \( (x-2)^2 \):** This factor is always non-negative and equals zero at \( x = 2 \). It doesn't change sign around \( x = 2 \).

Since each of these factors is squared, \( h'(x) \) is always non-negative and only equals zero at \( x = 0 \), \( x = 1 \), and \( x = 2 \).

### Step 3: Conclusion
For a relative maximum to occur, the derivative must change from positive to negative. However, \( h'(x) \) does not change sign at any of the critical points. Instead, it only touches zero without changing from positive to negative.

**Therefore, the graph of \( h(x) \) has no relative maximum points.**

Would you like more details, or do you have any questions?

Here are some related questions you might find helpful:
1. What are the conditions for a function to have a relative minimum?
2. How do you find inflection points using the second derivative?
3. Can a polynomial function have more relative minima than maxima?
4. How does the degree of the polynomial affect the number of critical points?
5. What is the significance of the multiplicity of roots in the derivative?

**Tip:** When dealing with squared factors in the derivative, always check if the sign of the derivative changes around the critical points to determine if there's a relative maximum or minimum.

Explore how to determine the number of relative maximum points for a polynomial function with a given derivative. This problem involves analyzing critical points and understanding the behavior of derivatives. Suitable for advanced high school students.

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