Optimization (Application in Calculus)

Question

An individual’s ABG oxygen level, L , at time t ∈ (0, π) , is given by L(t) = 90 +

sin(2t) + cos(2t). Find the values of the maximum and minimum oxygen level. When do these

values occur?

Answer 1

While you can use Calculus, you can also recognize that sin2t + cos2t = √2sin(2t+π/4), so L(t) = 90 + √2sin(2t+π/4).

sin's maximum is at π/2, where it equals 1, and its minimum is at 3π/2, where it equals -1

2t+π/4 = π/2; t = π/8. The maximum oxygen level is 90 + √2

2t+π/4 = 3π/2, t = 5π/8. The minimum oxygen level is 90 - √2

As sin equals 1 and -1 only once on [0, 2π], and we have 2t here, we have solutions every 2π/2 = π for the maximum and minimum; as we are looking for solutions on [0, π], these are our unique solutions.