The graph of f ′(x), the derivative of x, is continuous for all x and consists of five line segments as shown below. Given f (5) = 10, find the absolute minimum value of f(x) over the interval [0, 5].
please explain lol
To find the absolute minimum value of f(x) over the interval [0, 5], we need to analyze the information provided about the derivative function f ′(x) and use it to determine the behavior of the original function f(x).
From the information given, we know that the graph of f ′(x) consists of five line segments. Let's denote the points of intersection between these line segments as A, B, C, D, and E, from left to right.
Since f ′(x) is the derivative of f(x), the graph of f(x) will show the behavior of f ′(x) through its slopes.
To determine the behavior of f(x) over the interval [0, 5], we need to consider the following:
1. Slope before point A: Since f(x) is increasing, the slope of f(x) is positive for x < A.
2. Slope between points A and B: As the slope starts decreasing, it will become zero at point A and continue to be negative until point B. This indicates that f(x) reaches a local maximum at point A and a local minimum at point B.
3. Slope between points B and C: The slope becomes zero again at point B and remains zero until point C. This means that f(x) has another local maximum at point B.
4. Slope between points C and D: The slope becomes positive again at point C and remains positive until point D. This indicates that f(x) is increasing over this interval.
5. Slope between points D and E: The slope becomes zero at point D and remains zero until point E. Thus, f(x) has a local maximum at point D.
Based on the given conditions, we can infer that f(x) has an absolute minimum value over the interval [0, 5] at point B.
Therefore, the absolute minimum value of f(x) over the interval [0, 5] is the y-coordinate of point B on the graph of f(x).
To find the absolute minimum value of f(x) over the interval [0, 5], we need to examine the behavior of the derivative f'(x) and the given information about f(5) = 10.
The graph of f'(x) consists of five line segments, indicating that f(x) is piecewise linear. To find the points where the derivative changes its slope, we need to identify the critical points and endpoints of the interval [0, 5].
Since f'(x) is continuous for all x, the derivative changes its slope at the critical points and endpoints. Let's denote these points as C1, C2, C3, and C4.
Now, using the given information that f(5) = 10, we know that the function f(x) passes through the point (5, 10). This means that C4 is (5, 10).
To find the absolute minimum value of f(x), we need to examine the behavior of the derivative around the critical points and endpoints. We will compare the values of f(x) at these points to identify the lowest value.
1. Starting from the left endpoint x = 0, we evaluate the value of f(x) at x = 0.
2. Next, we evaluate the value of f(x) at C1.
3. Then, we evaluate the value of f(x) at C2.
4. After that, we evaluate the value of f(x) at C3.
5. Finally, we evaluate the value of f(x) at C4 (given as (5, 10)).
By comparing these values, you can determine the absolute minimum value of f(x) over the interval [0, 5].
Note: Since the graph of f'(x) consists of line segments, you might need to find the equations of these segments to perform the evaluations.
I can't see your graph, but google will provide numerous similar problems, such as
https://brainly.com/question/2497381