Question

If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in ______.

Options

• `(-oo, -1) ∪ (1, oo)`

• `[2, oo)`

• `(-oo, -1) ∪ [2, oo)`

• `(-oo, 0] ∪ (2, oo)`

Answer 1

If f(x) = `x^(3/2) (3x - 10)`, x ≥ 0, then f(x) is increasing in `[2, oo)`.

Explanation:

f(x) = `x^(3/2) (3x - 10)`, x ≥ 0

∴ f'(x) = `3/2 x^(1/2) (3x - 10) + x^(3/2) (3)`

= `15/2 x^(1/2) (x - 2)`

For f(x) to be increasing,

f'(x) ≥ 0 ⇒ `15/2 x^(1/2) (x - 2) ≥ 0`

⇒ x ≥ 2

⇒ x ∈ `[2, oo)`