Introduction

In the field of calculus, understanding the behavior of a function is essential to various applications, including optimization problems. Among the key points to consider are the local extrema, which include local maxima and minima. This article aims to walk through the process of finding and classifying the local extrema of the function (y x^3 - 6x^2 - 7).

Deriving the Critical Points

To find the critical points of the function, we start by computing its first derivative:

y x^3 - 6x^2 - 7

y' d/dx (x^3 - 6x^2 - 7) 3x^2 - 12x

Solving for Critical Points

Setting the first derivative equal to zero allows us to find the critical points:

3x^2 - 12x 0

3x(x - 4) 0

x 0 or x 4

The critical points of the function are (x 0) and (x 4).

Classifying the Extrema

Once we have the critical points, we can determine the intervals over which the function is either increasing or decreasing. This is done by testing the sign of the first derivative within these intervals.

Let's partition the domain into the following intervals: (-infty, -4, 0, 4, infty).

Interval Test Point Signed (f'(x)) Behavior of (f(x)) (-infty, -4) (-5) (15) Increasing (-4, 0) (-1) (-9) Decreasing (0, 4) (1) (15) Increasing (4, infty) (5) (15) Increasing

Behavior of the Function

Based on the intervals and signs of (f'(x)), we can classify the extrema:

(x -4): This point is where the function transitions from increasing to decreasing, indicating a local maximum. (x 0): This point is where the function transitions from decreasing to increasing, indicating a local minimum.

Conclusion

For the function (y x^3 - 6x^2 - 7), there is one local maximum at (x -4) and one local minimum at (x 0).

Graphical Representation and Visual Confirmation

Graphing the function provides a visual confirmation of the extrema:

Graph of (y x^3 - 6x^2 - 7)

The graph confirms that at (x -4), the function reaches a local maximum, and at (x 0), it reaches a local minimum.

Final Recap

The function (y x^3 - 6x^2 - 7) has a local maximum at (x -4). The function (y x^3 - 6x^2 - 7) has a local minimum at (x 0).