Have the following question and just wondering if my solution is correct(adsbygoogle = window.adsbygoogle || []).push({});

Let g(x)= x^5+3x-1. Show that there are no distinct points x_1, x_2 in R such that g(x_1)=g(x_2).

Proof by contradiction. Assume we have two solution x_1<x_2 in ℝ, i,e g(x_1)+g(x_2)=0, since g is differentiable on (x_1,x_2) and continuous on [x_1,x_2], then we can apply rolles theorem, there exits a C belonging to (x_1, x_2) such that df/dx=0, however df/dx=5x^4+3>0 Hence we have a contradiction and only one solution to f(x)=0.

Many thanks in advance.

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# Rolles Theorem, showing two distinct points.

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