How do I find local minimums based on f'(x) with line segments?
I have a picture of a graph with a the points (-4,0), (-3,2.5), (0,0), (2,2), and (5,0). Here is the question:
The graph of f ' (x), the derivative of f(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].
f' never crosses the x-axis so that means the function is always increasing, so is the minimum at x=0? And if it is, how do you find the f(x) value for x=0?
f'(x)=zero at x=-4,0,5. You are given max at x=5, so min has to be at x=0
now, f(x)=integral of f'(x) which is the area under the curve. So calculate the area under the curve from x=0 to 5, and subtract it from 10. In my head, I get about f(0)=6.5
To find the absolute minimum value of f(x) over the interval [0, 5] based on the given graph of f'(x), we need to first determine the critical points where f'(x) changes direction. Since f'(x) consists of five line segments, we can find these critical points by looking at the endpoints of each segment.
Let's analyze each segment one by one:
1. From (-4, 0) to (-3, 2.5): This segment has a positive slope, indicating that the derivative is increasing. Thus, f(x) is increasing in this interval.
2. From (-3, 2.5) to (0, 0): This segment has a negative slope, suggesting that the derivative is decreasing. Consequently, f(x) is decreasing in this interval.
3. From (0, 0) to (2, 2): This segment has a positive slope, implying that the derivative is increasing again. Meaning, f(x) is increasing in this interval.
4. From (2, 2) to (5, 0): This segment has a negative slope, indicating that the derivative is decreasing once more. Thus, f(x) is decreasing in this interval.
Now, let's focus on the interval [0, 5] and determine its absolute minimum.
Since f(x) is decreasing from 0 to 2 and then increasing from 2 to 5, the absolute minimum of f(x) on the interval [0, 5] can potentially occur at either the left endpoint (0) or a point of local minimum in the interval (2, 5).
To find the value of f(x) for x = 0, you can substitute x = 0 into the equation of f(x) based on the given graph or the points provided. From the points given, we know that f(0) = 0. So, the value of f(x) at x = 0 is 0.
However, to determine if there is a local minimum at any point within the interval (2, 5), we need additional information about the graph of f'(x) or the behavior of f(x) elsewhere. Without further details, we cannot definitively identify any local minimum within this specific interval.