Given that 𝑓(𝑥) = 3 cos 𝑥 + 2 sin 𝑥, where 0 ≤

𝑥 ≤ 90, for what value of 𝑥 𝑖𝑠 𝑓(𝑥) maximum?
A. 33.7
B. 56.3
C. 213.7
D. 236.3

1 answer

We can use calculus to find the maximum value of 𝑓(𝑥). Taking the derivative of 𝑓(𝑥), we get:

𝑓'(𝑥) = -3 sin 𝑥 + 2 cos 𝑥

Setting 𝑓'(𝑥) = 0 to find critical points, we get:

-3 sin 𝑥 + 2 cos 𝑥 = 0

Dividing both sides by cos 𝑥, we get:

-3 tan 𝑥 + 2 = 0

Solving for tan 𝑥, we get:

tan 𝑥 = 2/3

Using a calculator, we find that 𝑥 ≈ 33.7° or 𝑥 ≈ 213.7°. We can check these values to see which one gives us the maximum value of 𝑓(𝑥).

At 𝑥 = 33.7°, we have:

𝑓(33.7°) = 3 cos 33.7° + 2 sin 33.7° ≈ 3.29

At 𝑥 = 213.7°, we have:

𝑓(213.7°) = 3 cos 213.7° + 2 sin 213.7° ≈ -0.26

Therefore, the value of 𝑥 for which 𝑓(𝑥) is maximum is 𝑥 ≈ 33.7°, which is closest to option A.