Question

Prove f:R —> R, f (x) = 2ˣ+2⁻ˣ is increasing for x ∈ (0, ∞) and decreasing for x ∈ (-∞, 0)

Answer 1

we have to prove , f(x) = 2ˣ+2⁻ˣ is increasing for x ∈ ( 0, ∞) and decreasing for x ∈ ( -∞, 0)

concept : any function , y = f(x) is decreasing on (a, b) only if f'(x) < 0 and increasing only if f'(x) > 0 on (a, b).

so, first of all, differentiate f(x) with respect to x,

f'(x) = 2ˣln2 -2⁻ˣln2 = ln2(2ˣ - 2⁻ˣ)

putting, f'(x) = ln2(2ˣ - 2⁻ˣ) = 0

⇒2ˣ = 2⁻ˣ

⇒ x = 0

case 1 : when x > 0

⇒ 2ˣ > 2⁻ˣ

⇒2ˣ - 2⁻ˣ > 0

⇒ln2(2ˣ - 2⁻ˣ) > 0 ⇒f'(x) > 0

hence, f(x) is increasing for x ∈ (0, ∞)

case 2 : when x < 0

⇒2ˣ < 2⁻ˣ

⇒2ˣ - 2⁻ˣ < 0

⇒ln2(2ˣ - 2⁻ˣ) < 0 ⇒f'(x) < 0

hence, f(x) is decreasing for x ∈ (-∞, 0)