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 (0) = 6, find the absolute maximum value of f (x) over the interval [0, 3].
a) 0
b) 8
c) 10
d) 16
I mean 10
We are given graph of f'(x). f(0)=6
We need to find the absolute minimum value of f(x) over interval [0,3]
First we will see the graph of f'(x) over interval [0,3]
f'(3)=3
f'(0)=0
Thus, f'(x) is decreasing
x=0 is critical point of the function f(x) because f'(0)=0
We will get absolute maximum/minimum at x=0. f(x) >0
Hence, f(0) is absolute minimum at x=0 , Absolute minimum = 8
This might be right.
f(x) is increasing sorry not decreasing
well the answer is 8
To find the absolute maximum value of f(x) over the interval [0, 3], we need to analyze the graph of f'(x) and find its maximum points.
From the given information that the graph of f'(x) consists of five line segments, we can determine that the function f(x) is a piecewise linear function. Each line segment in the graph of f'(x) represents a constant slope segment in the graph of f(x).
Let's label the segments in the graph of f'(x) as follows:
Segment 1: slope = 2
Segment 2: slope = -1
Segment 3: slope = 0
Segment 4: slope = 3
Segment 5: slope = -2
Now, we need to determine the end points of each segment. Looking at the graph of f'(x), we can see that:
Segment 1 goes from x = 0 to x = 1
Segment 2 goes from x = 1 to x = 2
Segment 3 goes from x = 2 to x = 2.5
Segment 4 goes from x = 2.5 to x = 3
Segment 5 goes from x = 3 back to x = 0
Now, let's find the values of f(x) at the endpoints of each segment:
Segment 1:
f(0) = 6
Segment 2:
f(1) = f(0) + (slope of segment 1) * (length of segment 1)
f(1) = 6 + 2 * 1
f(1) = 8
Segment 3:
f(2) = f(1) + (slope of segment 2) * (length of segment 2)
f(2) = 8 + (-1) * 1
f(2) = 7
Segment 4:
f(2.5) = f(2) + (slope of segment 3) * (length of segment 3)
f(2.5) = 7 + 0 * 0.5
f(2.5) = 7
Segment 5:
f(3) = f(2.5) + (slope of segment 4) * (length of segment 4)
f(3) = 7 + 3 * 0.5
f(3) = 8.5
Finally, we need to check the value of f(x) at x = 0, which is the starting point. Comparing it with the previous values, we can see that f(0) = 6 is the absolute maximum value of f(x) over the interval [0, 3].
Therefore, the answer is a) 0.