Critical Points question Calc 1

Question

Find the critical numbers of the function. (Enter your answers as a comma-separated list.)

f(𝜃) = 10sec 𝜃 + 5tan 𝜃,    0 < 𝜃 < 2𝜋

Answer 1

f(θ) = 10secθ + 5tanθ, f is undefined at θ = π/2, 3π/2.

f'(θ) = 10secθtanθ + 5sec²θ = 5secθ(2tanθ + secθ) = 0

f'(θ) = 0 when 2tanθ = -secθ. So, 2(sinθ / cosθ) = -1 / cosθ Thus, sinθ = -1/2. So, θ = 7π/6 or 11π/6 are the critical numbers in the interval (0, 2π). Even though f'(θ) is undefined at θ = π/2 and 3π/2, π/2 and 3π/2 are not considered critical numbers of f(θ) as π/2 and 3π/2 are not in the domain of f(θ).

Answer 2

The critical numbers are where there first derivative equals 0.

The first derivative of the function f is f'(Θ)= 10sec(Θ)tan(Θ) + 5 sec²(Θ). This can be factored as

f'(Θ)= 5sec(Θ)[2tan(Θ)+sec(Θ)]. Graphing this function, we find the zero is at Θ = 7π/6. Therefore the critical value is Θ = 7π/6.