Find the absolute maximum and absolute minimum values of f on the given interval [π/4, 7π/4]. (Round all answers to two decimal places.)

f(t) = t + cot(t/2)

what's the problem?

f'(t) = 1 - 1/2 csc^2(t/2)

2 = csc^2(t/2)
√2 = csc(t/2)

csc pi/4,3pi/4 = √2
t/2 = pi/4 or 3pi/4
t = pi/2 or 3pi/2

Steve, thank you so much.

When plugging in the t-values to find the f(t)'s, I mistakenly put in the t/2 points.

You cleared that right up for me. I owe you big time. Karma points.

To find the absolute maximum and absolute minimum values of the function f(t) = t + cot(t/2) on the interval [π/4, 7π/4], we need to find the critical points and the endpoints of the interval, and then evaluate the function at these points.

Step 1: Find the derivative of f(t) with respect to t.
f'(t) = 1 - (1/2)csc^2(t/2)

Step 2: Set the derivative equal to zero to find critical points.
1 - (1/2)csc^2(t/2) = 0
csc^2(t/2) = 2
sin^2(t/2) = 1/2
sin(t/2) = ± √(1/2)
t/2 = arcsin(± √(1/2))
t = 2arcsin(± √(1/2))
t = π/2, 3π/2, 5π/2

Step 3: Evaluate the function f(t) at the critical points and the endpoints of the interval.
f(π/4) = π/4 + cot(π/8) ≈ 1.57
f(7π/4) = 7π/4 + cot(7π/8) ≈ -0.71
f(π/2) = π/2 + cot(π/4) ≈ 2.57
f(3π/2) = 3π/2 + cot(3π/4) ≈ 2.09
f(5π/2) = 5π/2 + cot(5π/4) ≈ -0.71

Step 4: Compare the function values to determine the absolute maximum and minimum.
The absolute maximum value is approximately 2.57 and occurs at t = π/2.
The absolute minimum value is approximately -0.71 and occurs at t = 7π/4 and t = 5π/2.

To find the absolute maximum and minimum values of the function f(t) = t + cot(t/2) on the given interval [π/4, 7π/4], we need to follow these steps:

1. Find the critical points of the function within the interval.
2. Evaluate the function at the critical points and the endpoints.
3. Compare the function values to determine the absolute maximum and minimum.

Step 1: Find the critical points
To find the critical points, we need to find where the derivative of the function is either zero or undefined.

The derivative of f(t) is obtained by using the product rule and the chain rule:

f'(t) = 1 - (1/2) * csc^2(t/2)

To find the values where f'(t) = 0, we need to solve the equation:

1 - (1/2) * csc^2(t/2) = 0

Simplifying the equation, we get:

csc^2(t/2) = 2

Taking the reciprocal of both sides, we have:

sin^2(t/2) = 1/2

Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite the equation as:

1 - cos^2(t/2) = 1/2

cos^2(t/2) = 1/2

Taking the square root of both sides, we get:

cos(t/2) = ±1/√2

Now, we solve for t/2:

t/2 = π/4 + nπ/2, or t/2 = 3π/4 + nπ/2

where n is an integer.

Step 2: Evaluate the function at the critical points and endpoints
a) Evaluate f(t) at the critical points t/2 = π/4 + nπ/2:

For t/2 = π/4, f(t) = π/4 + cot(π/8) = π/4 + √2
For t/2 = 3π/4, f(t) = 3π/4 + cot(3π/8) = 3π/4 - √2
For t/2 = 5π/4, f(t) = 5π/4 + cot(5π/8) = 5π/4 + √2
For t/2 = 7π/4, f(t) = 7π/4 + cot(7π/8) = 7π/4 - √2

b) Evaluate f(t) at the endpoints of the interval [π/4, 7π/4]:

For t = π/4, f(t) = π/4 + cot(π/8) = π/4 + √2
For t = 7π/4, f(t) = 7π/4 + cot(7π/8) = 7π/4 - √2

Step 3: Determine the absolute maximum and minimum
To find the absolute maximum and minimum values, we compare the function values obtained in Step 2.

The maximum value is the larger of the function values:
Maximum value = max{π/4 + √2, 3π/4 - √2, 5π/4 + √2, 7π/4 - √2}

The minimum value is the smaller of the function values:
Minimum value = min{π/4 + √2, 3π/4 - √2, 5π/4 + √2, 7π/4 - √2}

After evaluating these expressions, round the results to two decimal places to get the final values for the absolute maximum and minimum values of f on the given interval.