Question

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

Answer 1

Correct Answer: 2

Step 1: Transform the equation

We start by dividing through by \( t^2 \) to simplify the equation:

\[ \frac{e^{2x}}{t^2} = t^4 - t^3 - 3t^2 - t + 1 = 0 \]

Step 2: Substitute \( t + 1 \) and use the identity

We make the substitution \( t = u \) and transform the equation further:

\[ t^2 + 1 = t^2 - t + 1 - 3 = 0 \] which simplifies to the quadratic equation: \[ u^2 - u - 5 = 0 \]

Step 3: Solve for \( u \)

The quadratic equation \( u^2 - u - 5 = 0 \) has roots given by:

\[ u = \frac{1 \pm \sqrt{21}}{2} \]

Thus, the solutions for \( t \) are:

\[ t = 1 + \frac{\sqrt{21}}{2} \quad \text{or} \quad t = 1 - \frac{\sqrt{21}}{2} \]

Step 4: Conclude the solution

There are two real values of \( t \), corresponding to the two roots of the transformed equation.

Final Answer

There are two real values of \( t \).