Question

The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` ______.

Options

• increases in `[1/2, ∞)`

• decreases `(-∞, 1/2]`

• increases in (–∞, ∞)

• decreases in `[1/2, ∞)`

Answer 1

The function f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx` `underlinebb("increases in" [1/2, ∞))`.

Explanation:

Given: f(x) = `(4x^3 - 3x^2)/6 - 2sinx + (2x - 1)cosx`

Now, f(x) = (2x² – x) – 2cosx + 2cosx – sinx(2x – 1) = (2x – 1)(x – sinx)

for x > 0, x – sinx > 0

and x < 0, x – sin x < 0

for `x ∈ (-∞, 0] ∪ [1/2, ∞), f^'(x) ≥ 0` 

for `x ∈ [0, 1/2], f^'(x) ≤ 0`

⇒ f(x) increases in `[1/2, ∞)`.