To determine the difference between the maximum and minimum values that a function f(x) can take, we need to find the maximum and minimum values separately and then calculate the difference between them.

To find the maximum value of f(x), we need to find the highest point on the graph of the function. This can be done by finding the critical points where the derivative of the function is equal to zero or undefined. By analyzing the behavior of the function around these critical points, we can determine the maximum value.

Similarly, to find the minimum value of f(x), we need to find the lowest point on the graph of the function. Again, this can be done by finding the critical points and analyzing the behavior of the function.

Once we have identified the maximum and minimum values, we can calculate their difference.

Let's consider an example to illustrate this concept:

Suppose we have the function f(x) = x^2 - 4x + 3.

To find the critical points, we take the derivative of f(x) and set it equal to zero:

f'(x) = 2x - 4 = 0

Solving this equation, we find x = 2.

Now, we need to analyze the behavior of the function around this critical point.

For x < 2,="" the="" function="" is="" increasing,="" and="" for="" x="" /> 2, the function is decreasing. Therefore, the point (2, f(2)) is the maximum point on the graph.

To find the minimum value, we can analyze the behavior of the function as x approaches negative infinity or positive infinity. In this case, as x approaches negative infinity or positive infinity, the function increases without bound. Therefore, there is no minimum value for this function.

In this example, the maximum value of f(x) is 1, and there is no minimum value. Hence, the difference between the maximum and minimum values is 1.

In conclusion, the difference between the maximum and minimum values that f(x) can take is 1.