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Math Help - Show that x^2*e^|x|=2 has two solutions

  1. #1
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    Question Show that x^2*e^|x|=2 has two solutions

    How would I do this? I know from the graph that the roots are -0.9 and 0.9 but how do I show this??? Any help much appreciated.
    Thank you
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  2. #2
    Senior Member Dinkydoe's Avatar
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    If you may use that f(x) = x^2e^{|x|}-2 is continuous, it shouldn't be that hard: f[a,b]\to \mathbb{R} must reach all values in [f(a), f(b)].

    (look up theorem of Bolzano. I believe there's other even other names for that)

    given that f(1) = e-2 > 0 and f(1/2) < 0 it follows that there exists a x_0 with 1/2 < x_0 < 1 such that f(x_0) = 0.

    You can use this argument twice to show the existence of 2 roots (without necessarily finding them).
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  3. #3
    Senior Member Dinkydoe's Avatar
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    I meant the Intermediate value Theorem. It says: If f is contunious on a closed interval f:[a,b]\to \mathbb{R} then for any c\in [f(a),f(b)] exists a x_0\in [a,b] such that f(x_0)=c.
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