Question

Prove that f(x) = x² is increasing in interval (0, ∞) and decreasing in interval (-∞,0).

Answer 1

Let x₁, x₂ ∈ [0,∞] is such that

x₁ < x₂

∴ x₁ < x₂ ⇒ x₁²< x₁x₂ …..(i)

and x₁ < x₂ ⇒ x₁x₂ < x₂²……(i)

From (i) and (ii),

x₁ < x₂ ⇒ x₁² < X₂²

⇒ f(x₁) < f(x₂)

∴ X₁ < X₂ ⇒ f(x₁) < f(x₂)

where x₁, x₂ ∈ [0, ∞]

Hence, f(x) is continuously increasing in [0, ∞)

Again, let x₁, x₂ ∈ (-∞, 0) is such that

x₁ < x₂

Then

x₁ < x₂ ⇒ x₁²> x₁x₂

∵ – 3 < – 2, (- 3) (- 3) = 9

(- 3) × (- 2) = 6

∴ 9 > 6

x₁² > x₁ x₂

Again

x₁ < x₂ ⇒ x₁x₂ > x₂²

Again – 3 < – 2

(- 3) × (- 2) = 6

(- 2) × (- 2) = 4

6 > 4
∴ x₁x₂ > x₂²

From (i) and (ii),

x₁ < x₂ ⇒ X₁² > x₂²

⇒ f(x₁) > f(x₂)

∴ x₁ < x₂ ⇒ f(x₁) > f(x₂)

Hence, f(x) is continuously decreasing in (-∞, 0).