Final answer:
To find f² θ, square the given function fθ. To find the critical points of f θ on the interval (0 ≤ θ ≤ π), set the derivative of f θ equal to zero and solve for θ.
Step-by-step explanation:
To find f² θ, we need to square the given function fθ. Since the function fθ is given by fθ = 7 sin² θ - 7 cos θ, we can square it as f² θ = (7 sin² θ - 7 cos θ)². Expanding this expression, we get f² θ = 49 sin⁴ θ - 98 sin² θ cos θ + 49 cos² θ.
To find the critical points of f θ on the interval (0 ≤ θ ≤ π), we need to find the values of θ where the derivative of f θ is equal to zero. Let's find the derivative of f θ first. The derivative of f θ is f'(θ) = 28 sin³ θ - 28 sin θ cos² θ - 14 cos θ sin θ.
Next, we set f'(θ) equal to zero and solve for θ. 28 sin³ θ - 28 sin θ cos² θ - 14 cos θ sin θ = 0. We can factor out 14 sin θ from this equation, giving us 14 sin θ (2 sin² θ - cos² θ - 1) = 0. Now, we have two possibilities: either sin θ = 0 or 2 sin² θ - cos² θ - 1 = 0. From the first possibility, sin θ = 0 when θ = 0 or θ = π. From the second possibility, we can simplify the equation 2 sin² θ - cos² θ - 1 = 0 by using trigonometric identities. We know that sin² θ + cos² θ = 1, so we can substitute this into the equation and get 3 sin² θ - 1 = 0. Solving this equation, we find sin² θ = 1/3, which means sin θ = ±√(1/3). Therefore, the critical points of f θ on the interval (0 ≤ θ ≤ π) are θ = 0, π, arcsin(√(1/3)), and π - arcsin(√(1/3)).