Question

The values of b for which the function f(x) = cos x + bx+ a decreases on R are

• A. [−1, 1]
• B. (-∞, 1]
• C. (-∞, -1]
• D. (-1, 1)

Answer 1

The Correct Option is B

To determine for which values of b the function f(x) = cos x + bx + a decreases on ℝ, we need to analyze the derivative of the function. The function decreases when its derivative is less than or equal to zero.

1. First, find the derivative of f(x):
f'(x) = derivative of cos x + bx.
Using derivative rules, we get:

f'(x) = −sin x + b.

2. For f(x) to be decreasing, we need:
f'(x) ≤ 0 for all x ∈ ℝ.

This translates to: −sin x + b ≤ 0
Or equivalently: b ≤ sin x.

3. The range of sin x is [−1, 1] for all x ∈ ℝ, meaning the maximum value sin x can take is 1. Therefore:

b ≤ 1

4. Hence, for the function to be decreasing for all x on ℝ, the values of b must satisfy:

b ≤ 1, which is (-∞, 1].

This matches the correct option: (-∞, 1].