Find the absolute maximum and absolute minimum values of f on the given interval. (Round all answers to two decimal places.)
f(t) = t + cot(t/2)
We need the given interval
You're so right, my bad.
Interval: [π/4, 7π/4]
To find the absolute maximum and absolute minimum values of the function f(t) on the given interval, we will follow these steps:
1. Determine the critical points of f(t) by finding where the derivative is either zero or undefined.
2. Assess the endpoints of the interval.
3. Evaluate the function at each critical point and endpoint.
4. Compare the values obtained to find the absolute maximum and minimum.
Step 1: Finding the critical points
To find the critical points, we need to find where the derivative of f(t) is zero or undefined. Let's start by finding the derivative of f(t):
f'(t) = 1 - (1/2) * (1/sin^2(t/2))
Setting f'(t) equal to zero, we can solve the equation:
1 - (1/2) * (1/sin^2(t/2)) = 0
Simplifying the equation, we have:
1/sin^2(t/2) = 2
Taking the reciprocal of both sides:
sin^2(t/2) = 1/2
Then, taking the square root of both sides, we get:
sin(t/2) = 1/sqrt(2)
To find the values of t, we need to consider the domain of the inverse sine function (also known as arcsin) and use the reference angle:
arcsin(1/sqrt(2)) = pi/4
Since sine is positive in the first and second quadrants, we have:
t/2 = pi/4 and t/2 = pi - pi/4
Solving for t, we get:
t = pi/2 and t = 3pi/2
Therefore, t = pi/2 and t = 3pi/2 are the critical points of f(t).
Step 2: Assessing the endpoints
Since we were not given the interval, we need to determine the interval for which we want to find the absolute maximum and minimum. Once we have the interval, we can assess if the endpoints are critical points or not.
Step 3: Evaluating the function at critical points and endpoints
Now that we have the critical points, we can evaluate the function f(t) at these points and at the endpoints of the interval to find the corresponding function values.
f(pi/2) = (pi/2) + cot(pi/4) = (pi/2) + 1 = (pi + 2)/2
f(3pi/2) = (3pi/2) + cot(3pi/4) = (3pi/2) - 1 = (3pi - 2)/2
Step 4: Comparing to find the absolute maximum and minimum
Now that we have the values of f(t) at the critical points and endpoints, we compare them to find the absolute maximum and minimum.
Comparing the values:
f(pi/2) = (pi + 2)/2
f(3pi/2) = (3pi - 2)/2
Therefore, (pi + 2)/2 is the absolute maximum value, and (3pi - 2)/2 is the absolute minimum value of f(t) on the given interval.
In summary, the absolute maximum value of f(t) is (pi + 2)/2, and the absolute minimum value is (3pi - 2)/2 on the given interval.