The number of real roots of the equation e 4 x + 2 e 3 x - e x - 6 = 0 is :

Step-by-step solution

Given that e 4 x + 2 e 3 x - e x - 6 = 0 Let    e x = t , t > 0 f ( t ) = t 4 + 2 t 3 - t - 6 f ' ( t ) = 4 t 3 + 6 t 2 - 1 f ' ' ( t ) = 12 t 2 + 12 t > 0 ( t > 0 ) f ' ( t ) is increasing function Now f ' ( 0 ) = - 1 f ' ( 1 ) = 9 By using Intermediate value theorem there exists a root for f x   =   0   if f α f β < 0   ⇒   f γ   =   0   ,   γ ∈   α ,   β So f ' (