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  4. The number of real roots of the equation e4 x+2 e3 x-ex-6=0 is :

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Q. The number of real roots of the equation $e^{4 x}+2 e^{3 x}-e^{x}-6=0$ is :

JEE MainJEE Main 2021Application of Derivatives

Solution:

Let $e ^{ x }= t >0$
$f(t)=t^{4}+2 t^{3}-t-6=0 $
$f'(t)=4 t^{3}+6 t^{2}-1$
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$f''( t )=12 t ^{2}+12 t >0$
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$f(0)=-6, f(1)=-4, f(2)=24$
$\Rightarrow $ Number of real roots $=1$