Find the number of solutions of this equation?

#|x|e^|x|=1#

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1 Answer
George C.
Apr 11, 2018

#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]}

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