Question

. 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

Answer 1

We can use calculus to find the maximum value of 𝑓(𝑥). First, we take the derivative of 𝑓(𝑥) with respect to 𝑥:

𝑓′(𝑥) = −3 sin 𝑥 + 2 cos 𝑥

To find the maximum value of 𝑓(𝑥), we need to set 𝑓′(𝑥) equal to zero and solve for 𝑥:

−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°

Since the given range for 𝑥 is 0 ≤ 𝑥 ≤ 90, the only possible answer is A. 33.7. We can confirm that this is the correct answer by checking that 𝑓(33.7) is indeed greater than 𝑓(0) and 𝑓(90).