Question

The equation $x+e^x=0$ has

• A. only one real root
• B. only two real roots
• C. no real root
• D. none of these

Answer 1

The correct option is A only one real root

Let $f\left(x\right)=x+e^x=0$.

Since $f(-\infty )=-\infty$ and $f(+\infty )=\infty$,

$\therefore f\left(x\right)=0$ has a real root.

Let the real root be $\alpha$. Then $f\left(\alpha \right)=0$.

Now, $f^{\prime }\left(x\right)=1+e^x>0,\forall x\in R$

$\therefore f\left(x\right)$ is an increasing function $\forall x\in R$.

$\therefore$ for any other real number $\beta$

$f\left(\beta \right)>f\left(\alpha \right)$ or $f\left(\beta \right)<f\left(\alpha \right)$.

$\therefore f\left(x\right)=0$ has no other real root.

Hence, the equation has only one real root.