Question

Find the interval for which the function f (x) = e – ax + eax, a > 0 is monotonically decreasing.

Answer 1

Given, f(x) = e– ax + eax, a > 0

⇒ f'(x) = –ae–ax + aeax

⇒ f'(x) = a(eax – e– ax)

Since f(x) is monotonic increasing, so f'(x) > 0

⇒ a(eax – e–ax) > 0

⇒ a((e2ax – 1)/eax) > 0

⇒ e2ax – 1 > 0

⇒ e2ax > e0

⇒ 2ax > 0

⇒ x > 0

Hence, the required interval (0, ∞).