Question

If f(x) = 2e^x − ae^-x + (2a + 1)x –3 is increasing for all values of x then

• A. a ∈ (-∞, ∞)
• B. a ∈ (-∞, 0]
• C. a ∈ [0, ∞)
• D. None of these

Answer 1

Answer: (C)

a ∈ [0, ∞)

Answer 2

f '(x) = 2e^x + ae^-x + 2a + 1

= e^-x(2e^2x + (2a + 1)e* + a)

= $2 e ^ { - x } \left( e ^ { 2 x } + \left( a + \frac { 1 } { 2 } \right) e ^ { x } + \frac { a } { 2 } \right)$

= 2e^-x (e^x + a) $\left( e ^ { x } + \frac { 1 } { 2 } \right)$

For f(x) to be increasing f '(x) ≥ 0 ∀ x ∈ R ⇒ e^x + a ≥ 0 ∀ x ∈ R ⇒ a ≥ 0.