Find the absolute maximum and absolute minimum of f(\theta )=1 + cos^(2)(\theta ) on the interval [(\pi )/(4,\pi )]

Question

Find the absolute maximum and absolute minimum of f(\theta )=1 + cos^(2)(\theta ) on the interval [(\pi )/(4,\pi )]

Answer 1

There might be some typos in the statement of the problem.

Let me state my best guess of it and solve it.

Even if my guess is wrong, hope that it will help.

Find the absolute maxima and minima of f(θ) = 1 + cos²(θ) on the interval [π/4, π].

Minima and maxima occur at domain boundary or at a critical point.

In the case of f(θ), critical points are those where the derivative is 0.

So we need to compute the derivative, set it to 0, and find the solutions θ.

Let me just write down the derivative, assuming you know how to compute it.

f'(θ) = -2cos(θ)sin(θ) = -sin(2θ)

Find all θ where f'(θ) = 0:

-sin(2θ) = 0

2θ = kπ (for any integer k)

θ = kπ/2 (for any integer k)

When limited to the interval [π/4, π], the critical points are at θ = π/2 and π.

Including the boundary points we then have three candidates for maxima and minima.

We have to evaluate the function at all three and compare.

f(π/4) = 1 + cos²(π/4) = 3/2

f(π/2) = 1 + cos²(π/2) = 1

f(π) = 1 + cos²(π) = 2

Thus there is an absolute minimum at π/2 and absolute maximum at π.