How to check for fixed points in such type of questions
closed as off-topic by GNUSupporter 8964民主女神 地下教會, cansomeonehelpmeout, Xander Henderson, José Carlos Santos, Connor Harris May 11 '18 at 22:25
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Hint. If $f$ has two distinct fixed points $x_1$ and $x_2$ in $[a,b]$, then by the Mean Value Theorem, there is $t\in (a,b)$ such that $$x_1-x_2=f(x_1)-f(x_2)=f'(t)(x_1-x_2).$$
Moreover see Proof for the property of a fixed point of $f$.