Question

Find the number of solutions of this equation?

`|x|e^|x|=1`

Answer 1

`2` real solutions

Explanation:

Given:

`abs(x)e^abs(x) = 1`

Note that if `x != 0` is a solution, then so is `-x`.

Considering positive values of `x`, note that both `x` and `e^x` are strictly monotonically increasing functions, so `xe^x` is also strictly monotonically increasing.

Also note that:

`(color(blue)(1/2))e^(color(blue)(1/2)) < 1/2sqrt(4) = 1`

`(color(blue)(1))e^(color(blue)(1)) = e > 1`

So by the intermediate value theorem, there is some `x` in `(1/2, 1)` such that `xe^x = 1`

Then `-x` is also a solution of the given equation.

graph{(y-abs(x)e^abs(x))(y-1) = 0 [-2.802, 2.198, -0.57, 1.93]}