Question 9

The number of real roots of the equation

$$(e^x + e^{-x})^3 + 3(e^x + e^{-x})^2 + 3(e^x + e^{-x}) = 7$$

Solution

Let $$e^x+e^{-x}=P$$

$$P^3+3P^2+3P-7 = 0$$

$$\left(P-1\right)(P^2+4P+7)=0$$

P = 1 and the other two roots are imaginary roots

$$e^x + e^{-x} = 1$$ but the minimum value of $$e^x + e^{-x} is 2$$, as $$e^x$$ is a positive number and sum of a positive number and its reciprocal is at least 2.

No. of real roots is 0.

Hence A is the correct answer.

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PGDBA 2019

The number of real roots of the equation $$(e^x + e^{-x})^3 + 3(e^x + e^{-x})^2 + 3(e^x + e^{-x}) = 7$$

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The number of real roots of the equation

$$(e^x + e^{-x})^3 + 3(e^x + e^{-x})^2 + 3(e^x + e^{-x}) = 7$$