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The critical points of the function \( f(x) = \frac{x^4}{(x-1)(x-3)^3} \) are:

- \( x = 0 \)
- \( x = \frac{6}{5} \)

These critical points are where the function's slope is zero or undefined, but we need to evaluate the second derivative or test nearby points to classify them as maxima, minima, or saddle points. Would you like me to proceed with that analysis?

Here are 5 follow-up questions that might help:

1. How do we classify critical points using the second derivative test?
2. How can we check for concavity at critical points?
3. What happens to the function at \( x = 1 \) and \( x = 3 \) (where the denominator could be zero)?
4. How do we find the inflection points for this function?
5. What is the significance of higher-order critical points?

**Tip:** Critical points where the denominator is zero may indicate vertical asymptotes. These points are important for determining the behavior of the function.

Solve the maxima and minima points for the objective function f(x) = x^4 / ((x - 1)(x - 3)^3)

This page covers how to find the maxima and minima points of the function f(x) = x^4 / ((x - 1)(x - 3)^3). It explains critical points, the first derivative test, and the second derivative test, and is suitable for advanced high school students.

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