The number of real roots of the equation

Question:

The number of real roots of the equation

$e^{t x}+2 e^{3 x}-e^{x}-6=0$ is :

  1. 2

  2. 4

  3. 1

  4. 0


Correct Option: , 3

Solution:

Let $\mathrm{e}^{\mathrm{x}}=\mathrm{t}>0$

$f(t)=t^{4}+2 t^{3}-t-6=0$

$f^{\prime}(t)=4 t^{3}+6 t^{2}-1$

$f^{\prime \prime}(\mathrm{t})=12 \mathrm{t}^{2}+12 \mathrm{t}>0$

$f(0)=-6, f(1)=-4, f(2)=24$

$\Rightarrow$ Number of real roots $=1$

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