Finding Relative Maxima and Minima of the Function f(x) x^4 - 12x^3

In this article, we will explore the process of finding the relative maxima and minima of the function f(x) x^4 - 12x^3. This involves several steps including finding the first and second derivatives, setting up and solving equations, and analyzing the results. Let's dive into the detailed analysis.

Step 1: Find the First Derivative

The first step in finding the relative extrema is to find the first derivative of the function.

f(x) x^4 - 12x^3

f'(x) 4x^3 - 36x^2

Step 2: Set the First Derivative to Zero to Find Critical Points

The next step is to set the first derivative equal to zero and solve for the critical points.

4x^3 - 36x^2 0

Factor out the common terms:

4x^2(x - 9) 0

This gives us the critical points:

4x^2 0 rarr; x 0 x - 9 0 rarr; x 9

Thus, the critical points are:

x 0 x 9

Step 3: Determine the Second Derivative to Classify the Critical Points

The second step is to determine the second derivative to classify the critical points.

f''(x) 12x^2 - 72x

Classifying the Critical Point at x 0

First, we evaluate the second derivative at x 0:

f''(0) 12(0)^2 - 72(0) 0

Since f''(0) 0, we need to further investigate the behavior of the function around x 0.

Consider the first derivative f'(x) 4x^2(x - 9) around x 0: For x 0, f'(x) 0 For x 0, f'(x) 0

This indicates that the function is decreasing around x 0, suggesting that x 0 is not a maximum or minimum point but an inflection point.

Classifying the Critical Point at x 9

Next, we evaluate the second derivative at x 9:

f''(9) 12(9)^2 - 72(9) 972 - 648 324

Since f''(9) 0, the function has a local minimum at x 9.

Function Values at Extrema

Finally, we find the function values at these points:

f(0) 0^4 - 12(0)^3 0 f(9) 9^4 - 12(9)^3 6561 - 11664 -5123

Therefore, the local maximum and minimum are:

Summary of Relative Extrema

Local Maximum: (0, 0) Local Minimum: (9, -5123)

By following these steps, we have successfully found the relative extrema of the function f(x) x^4 - 12x^3.

Conclusion

In conclusion, understanding the process of finding relative extrema is crucial in calculus and function analysis. By following the steps of finding the first and second derivatives, setting them to zero, and analyzing the results, we can accurately identify the points of local maxima and minima. This knowledge is invaluable for a wide range of applications in mathematics and related fields.

Further Reading

If you enjoyed this article, you might also find these resources helpful:

Khan Academy: Second Derivative Test Introduction Lamar University: Local Extrema and Optimization