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

[AI-generated answer — Bot]

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.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find the value of 𝑥 for which 𝑓(𝑥) is maximum, we need to find the maximum value of the function 𝑓(𝑥) = 3 cos 𝑥 + 2 sin 𝑥.

𝑓(𝑥) = 3 cos 𝑥 + 2 sin 𝑥 can be written as 𝑓(𝑥) = 𝑟 sin(𝑥 + 𝜃), where 𝑟 is the magnitude of the vector and 𝜃 is the angle.

We can rewrite 𝑓(𝑥) = 3 cos 𝑥 + 2 sin 𝑥 as 𝑓(𝑥) = 3 (cos 𝑥/√(𝑡^2 + 3^2)) + 2 (sin 𝑥/√(𝑡^2 + 2^2)), where 𝑡 = tan(𝜃/2).

The maximum value of the function 𝑓(𝑥) occurs when 𝑡 is maximum, which means when 𝜃 is maximum.

The maximum value of 𝑡 is given by 𝑡 = tan(𝜃_max/2) = √(9/4) = 3/2.

To find the corresponding angle 𝜃_max, we use the inverse tangent function: 𝜃_max = 2 * atan(3/2).

𝜃_max ≈ 56.31 degrees.

Therefore, for 𝑥 = 𝜃_max, 𝑓(𝑥) is maximum.

The answer is B. 56.3.