Let g(s)=1/(s-2)on the interval (0,1). Find the absolute maximum and absolute minimum of g(s) on this interval
g'(s) = -1/(s-2)^2
setting this equal to zero has no solution, so the function has no max/min
and since g'(s) is always negative, the function is decreasing for all values of s except s = 2
so let's look at the endpoints of the interval
g(0) = 1/-2 = - 1/2
g(1) = 1/-1 = -1
so for the given interval, the max is -1/2 and the min is -1
To find the absolute maximum and absolute minimum of g(s) on the interval (0, 1), we need to evaluate the function at the critical points and endpoints, and compare their values.
Step 1: Calculate the derivative of g(s).
g'(s) = -1/(s-2)^2
Step 2: Find the critical points.
To find the critical points, we set the derivative equal to zero and solve for s.
-1/(s-2)^2 = 0
Since the denominator cannot be zero, there are no critical points in the interval (0, 1).
Step 3: Evaluate the function at the endpoints of the interval.
g(s) = 1/(s-2)
At s = 0:
g(0) = 1/(0-2)
= -1/2
At s = 1:
g(1) = 1/(1-2)
= 1
Step 4: Compare the values.
The function has an absolute maximum value of 1 at s = 1 and an absolute minimum value of -1/2 at s = 0 on the interval (0, 1).
Therefore, the absolute maximum of g(s) on the interval (0, 1) is 1, and the absolute minimum is -1/2.
To find the absolute maximum and minimum of the function g(s) = 1/(s-2) on the interval (0,1), we need to evaluate the function at its critical points and its endpoints.
1. Critical points: These are the points where the derivative of the function is either zero or undefined. In this case, the function g(s) is undefined at s = 2 because the denominator becomes zero. However, since the interval we are considering is (0,1), which does not include s = 2, we don't have any critical points within this interval.
2. Endpoints: The interval (0,1) has two endpoints, s = 0 and s = 1. We need to evaluate g(s) at these points to see if they represent the absolute maximum or minimum.
a) g(0) = 1/(0-2) = -1/2
b) g(1) = 1/(1-2) = -1
Now, to determine the absolute maximum and minimum, we compare these function values.
The absolute maximum occurs at s = 0, where g(s) = -1/2.
The absolute minimum occurs at s = 1, where g(s) = -1.
Therefore, the absolute maximum of g(s) on the interval (0,1) is -1/2, and the absolute minimum is -1.