Comparing Minimum Y Values Of F(x) And G(x) A Detailed Analysis

To determine which function, f(x) = 3|x-2| - 4 or g(x), has the lowest minimum y-value, we need to analyze each function separately. We are given the function f(x) explicitly and a table representing the derivative g'(x) of the function g(x). By understanding the properties of absolute value functions and the relationship between a function's derivative and its minimum values, we can accurately compare the minimum y-values of these two functions.

Analyzing f(x) = 3|x-2| - 4

Let's delve into the characteristics of the function f(x) = 3|x-2| - 4. This is an absolute value function, which is a transformation of the basic absolute value function |x|. Absolute value functions are known for their V-shaped graphs, with a distinct vertex representing either a minimum or a maximum point. The general form of an absolute value function is a|x-h| + k, where (h, k) is the vertex of the graph. The coefficient a determines the direction and steepness of the V-shape.

In our case, f(x) = 3|x-2| - 4, we can identify a = 3, h = 2, and k = -4. This tells us that the vertex of the graph is at the point (2, -4). Since a = 3 is positive, the V-shape opens upwards, indicating that the vertex represents the minimum point of the function. Therefore, the minimum y-value of f(x) is -4. Understanding this vertex form is crucial for quickly identifying the minimum or maximum value of any absolute value function.

Furthermore, the transformation |x-2| shifts the basic absolute value function 2 units to the right, and the term 3|x-2| vertically stretches the graph by a factor of 3. Finally, subtracting 4, i.e., -4, shifts the entire graph downwards by 4 units. This step-by-step breakdown of transformations helps visualize and comprehend how the function's graph is formed and, consequently, where its minimum value lies. The minimum value is a critical feature of absolute value functions, often used in various applications, including optimization problems.

Analyzing g(x) Using g'(x) Table

Now, let's examine the function g(x) using the information provided in the table of its derivative, g'(x). The derivative of a function, denoted as g'(x), provides crucial information about the function's slope and behavior. Specifically, the sign of the derivative indicates whether the function is increasing or decreasing. When g'(x) is negative, g(x) is decreasing, and when g'(x) is positive, g(x) is increasing. Points where g'(x) changes sign are critical points, which can indicate local minima or maxima.

We have the following data for g'(x):

From the table, we can see that g'(x) changes sign at two points: between x = -3 and x = -2, and between x = 0 and x = 1. Let's analyze these intervals:

1. Between x = -3 and x = -2: g'(x) changes from positive (2) to negative (-1). This indicates that g(x) is increasing before x = -2 and decreasing after x = -2. Thus, there is a local maximum at x = -2.
2. Between x = -1 and x = 1: g'(x) is negative at x = -1 (g'(-1) = -2), negative at x = 0 (g'(0) = -1), and becomes positive at x = 1 (g'(1) = 2). This shows that g(x) decreases until some point between x = 0 and x = 1 and then increases. This suggests that there's a minimum in the interval [0,1].

To approximate the minimum value of g(x), we need to consider the y-values corresponding to the x-values where g'(x) changes from negative to positive. However, the table only provides the values of g'(x), not g(x) itself. Without additional information or the actual function g(x), we cannot determine the exact y-value of the minimum. We can only infer that the minimum occurs somewhere between x = 0 and x = 1. The crucial understanding here is that the sign change in the derivative is a key indicator of local extrema.

Comparing the Minimum Values

Now, let's compare the minimum y-values of the two functions. We found that the minimum y-value of f(x) = 3|x-2| - 4 is -4. For g(x), we determined that a minimum occurs between x = 0 and x = 1, but we do not know the exact y-value at this minimum point. Without the specific y-value for the minimum of g(x), we cannot definitively say whether it is lower, higher, or the same as the minimum of f(x).

However, we can analyze the information available to make a reasoned comparison. The derivative values for g(x) provide insight into its behavior, but they do not directly give us the y-values. We know g(x) decreases until some point between x = 0 and x = 1 and then increases. To accurately compare the minima, we would need more information about g(x), such as its explicit form or the y-values at specific points.

In summary, we have a concrete minimum y-value of -4 for f(x), while the minimum y-value for g(x) is unknown but lies somewhere near the interval where its derivative changes from negative to positive. Given the information, we cannot definitively determine which function has the lowest minimum y-value.

Conclusion

In conclusion, we analyzed the function f(x) = 3|x-2| - 4 and determined its minimum y-value to be -4. For the function g(x), we used the table of its derivative g'(x) to identify a local minimum between x = 0 and x = 1. However, without additional information about g(x), such as its explicit formula or specific y-values, we cannot definitively compare its minimum y-value to that of f(x). Therefore, we cannot definitively say which function has the lowest minimum y-value based solely on the information provided. Comparing the behavior of functions through their derivatives and understanding transformations of absolute value functions are essential skills in mathematical analysis.