The number of points, where the curve f ( x ) = e 8 x - e 6 x - 3 e 4 x - e 2 x + 1 , x ∈ ℝ cuts x -axis, is equal to............

Step-by-step solution

Given, e 8 x - e 6 x - 3 e 4 x - e 2 x + 1 = 0 ⇒ e 4 x - e 2 x - 3 - 1 e 2 x + 1 e 4 x = 0 ⇒ e 4 x + 1 e 4 x + 2 - e 2 x + 1 e 2 x = 5 ⇒ e 2 x + 1 e 2 x 2 - e 2 x + 1 e 2 x = 3 Now let e 2 x + 1 e 2 x = t So, the equation becomes t 2 - t - 5 = 0 ⇒ t = 1 + 21 2 , ignoring negative sign as exponential function are positive, Now e 2 x + 1 e 2 x = 1 + 21 2 ⇒ e 4 x - 1 + 21 2 e 2 x + 1 = 0 which is quadratic equation in e 2 x with upward parabola, Now by A . M ≥ G . M we get, e 2 x +