de menor a mayor los valores criticos de f son

To find the critical points of the function \( f(x) \), we need to set the first derivative \( f'(x) \) to zero and solve for \( x \).

Given the first derivative:
\[ f'(x) = 48x^3 - 672x^2 + 2160x \]

We set \( f'(x) = 0 \):
\[ 48x^3 - 672x^2 + 2160x = 0 \]

Factor out the greatest common divisor (GCD):
\[ 48x(x^2 - 14x + 45) = 0 \]

This gives us one root directly:
\[ x = 0 \]

Next, we solve the quadratic equation:
\[ x^2 - 14x + 45 = 0 \]

We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ a = 1, \, b = -14, \, c = 45 \]
\[ x = \frac{14 \pm \sqrt{196 - 180}}{2} \]
\[ x = \frac{14 \pm \sqrt{16}}{2} \]
\[ x = \frac{14 \pm 4}{2} \]
\[ x = \frac{14 + 4}{2} \quad \text{and} \quad x = \frac{14 - 4}{2} \]
\[ x = 9 \quad \text{and} \quad x = 5 \]

So the critical points are:
\[ x = 0, \, x = 5, \, x = 9 \]

Ordered from smallest to largest, the critical points of \( f(x) \) are:
\[ 0, 5, 9 \]

Would you like more details or have any questions?

Here are 5 numbered questions you can ask next:
1. How do you find the local maxima and minima using the first derivative test?
2. How do you determine the concavity at the critical points?
3. What are the steps to find the inflection points of a function?
4. Can you explain the significance of the critical points in the graph of a function?
5. How do you solve higher-degree polynomial equations?

**Tip:** When solving for critical points, always check for common factors first to simplify the equation before solving the remaining polynomial.

Learn how to find critical points of the function f(x) = 48x^3 - 672x^2 + 2160x. This problem involves calculus, specifically finding roots of a polynomial derivative and using the quadratic formula. Suitable for advanced high school students.

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