Question

If f(x) = e^2x – (a + 1)e^x + 2x is increasing for all real values of x in R, then find the maximum value of a.

Answer 1

Given f(x) = e^2x – (a + 1)e^x + 2x

⇒ f'(x) = 2e^2x – (a + 1)e^x + 2

Since f(x) is increasing, so

2e^2x – (a + 1)e^x + 2 ≥ 0

Thus, its descriminant D < 0

⇒ (a + 1)² – 16 < 0

⇒ (a + 1)² – 4² < 0

⇒ (a + 1 + 4)(a + 1 – 4) < 0

⇒ (a + 5)(a – 3) < 0

⇒ –5 < a < 3

Hence, the maximum value of a is 3.