Determining Relative Extrema of the Function f(x) x^4 - 8x^3 18x^2

The process of finding the relative extrema of a function involves several key steps. In this article, we will explore how to determine the relative extrema of the function f(x) x^4 - 8x^3 18x^2 by finding and analyzing its critical points and using both the second derivative test and the first derivative test.

Step 1: Finding the First Derivative

The first derivative of a function provides us with its rate of change at any point. For the function f(x) x^4 - 8x^3 18x^2, we start by finding its first derivative:

f'(x) 4x^3 - 24x^2 - 36x

Step 2: Setting the First Derivative to Zero

To find the critical points, we need to set the first derivative equal to zero and solve for x:

4x^3 - 24x^2 - 36x 0

Factoring out the common term, we get:

4x(x^2 - 6x - 9) 0

This gives us:

4x 0 or x^2 - 6x - 9 0

Solving 4x 0 gives us:

x 0

For the quadratic x^2 - 6x - 9 0, we can factor it as:

(x - 3)^2 0

Therefore, we have a double root at:

x -3

Step 3: Finding the Second Derivative

The second derivative is used to determine the concavity of the function and to confirm the nature of the critical points:

f''(x) 12x^2 - 48x - 36

Step 4: Evaluating the Second Derivative at the Critical Points

Let's evaluate the second derivative at the critical points:

For x 0:

f''(0) 12(0)^2 - 48(0) - 36 36

Since the second derivative is positive at x 0, this indicates a local minimum.

For x -3:

f''(-3) 12(-3)^2 - 48(-3) - 36 108 144 - 36 216 - 36 180

The second derivative is positive at x -3, but we need to use the first derivative test to determine its nature. We need to check the sign of the first derivative around x -3:

Step 5: First Derivative Test for x -3

We choose test points around x -3:

For x -4:

f'(-4) 4(-4)^3 - 24(-4)^2 - 36(-4) -256 - 384 144 -592 144 -448

For x -2:

f'(-2) 4(-2)^3 - 24(-2)^2 - 36(-2) -32 - 96 72 -160 72 -88

Since the first derivative is negative on both sides of x -3, the function is decreasing on both sides, indicating that x -3 is a local maximum.

Summary of Relative Extrema

From the analysis:

Local minimum at (0, 0) Local maximum at (-3, 27)

These points are relative or local extremes of the function, indicating the lowest and highest points in the interval around these coordinates.