The number of real roots of the equation $e^{6x} - e^{4x} - 2e^{3x} - 12e^{2x} + e^x + 1 = 0$ is :
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For transcendental equations like $P(e^x)=0$, the number of real roots for $x$ is equal to the number of *positive* roots of the polynomial $P(t)=0$. Descartes' rule of signs is a very powerful tool to find an upper bound on the number of roots quickly.
Step 1: Understanding the Concept:
This equation involves exponential terms. We can transform it into a polynomial by substituting a new variable.
Since \(e^x\) is always positive for real \(x\), we are interested in the number of positive real roots of the resulting polynomial. Step 2: Key Formula or Approach:
Let \(e^x = t\), where \(t \in (0, \infty)\).
The equation becomes: \(t^6 - t^4 - 2t^3 - 12t^2 + t + 1 = 0\).
We analyze the function \(f(t) = t^6 - t^4 - 2t^3 - 12t^2 + t + 1\) for \(t>0\) using the Intermediate Value Theorem (IVT) and derivatives. Step 3: Detailed Explanation:
Evaluate \(f(t)\) at some specific values of \(t\):
\(f(0) = 0 - 0 - 0 - 0 + 0 + 1 = 1>0\).
\(f(1) = 1 - 1 - 2 - 12 + 1 + 1 = -12<0\).
Since the sign changes between \(t=0\) and \(t=1\), there exists at least one root \(t_1\) in the interval \((0, 1)\).
Now check for larger values of \(t\):
\(f(2) = 64 - 16 - 16 - 48 + 2 + 1 = -13<0\).
\(f(3) = 729 - 81 - 54 - 108 + 3 + 1 = 490>0\).
Since the sign changes between \(t=2\) and \(t=3\), there exists another root \(t_2\) in the interval \((2, 3)\).
Analyzing the derivative \(f'(t) = 6t^5 - 4t^3 - 6t^2 - 24t + 1\):
At very large \(t\), \(f(t)\) is dominated by \(t^6\), so it goes to \(\infty\).
The sign changes indicate that the function crosses the x-axis twice in the positive domain.
Checking for more roots using Descartes' Rule of Signs:
The coefficients are \(+1, -1, -2, -12, +1, +1\).
There are 2 sign changes (\(+1 \to -1\) and \(-12 \to +1\)), indicating at most 2 positive real roots.
Since we found 2 intervals with sign changes, there are exactly 2 positive roots for \(t\).
Each positive \(t\) corresponds to exactly one real \(x\) (since \(x = \ln t\)). Step 4: Final Answer:
The number of real roots is 2.
