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............

The curve f(x) = e^8x - e^6x - 3e^4x - e^2x + 1 cuts the x-axis where f(x) = 0.

So, we need to solve the equation e^8x - e^6x - 3e^4x - e^2x + 1 = 0.

This equation is a quadratic in terms of e^4x. Let's substitute e^4x = y. Then the equation becomes:

y^2 - y - 3y - 1 + 1 = 0

Simplifying, we get:

y^2 - 4y = 0

This equation has two solutions: y = 0 and y = 4.

Substituting back y = e^4x, we get two equations: e^4x = 0 and e^4x = 4.

The equation e^4x = 0 has no solution because the exponential function is always positive.

The equation e^4x = 4 can be rewritten as 4x = ln(4), which gives x = ln(4)/4.

So, the curve f(x) = e^8x - e^6x - 3e^4x - e^2x + 1 cuts the x-axis at one point: x = ln(4)/4.