For any real number $c$, the polynomial $x^3 + x + c$ has exactly one real root For any real number $c$, the polynomial $x^3 + x + c$ has exactly one real root.  
how can i show that above statement is true or false.can anyone help me please

Here’s yet another argument, not using calculus. As @xen remarks, a strictly increasing real-valued function will have at most one root. Letting $f$ be our function, we find that $f(x+\varepsilon)-f(x)=\varepsilon\lbrack 3x^2+3x\varepsilon+\varepsilon^2+1\rbrack$, and the quantity in brackets is a sum of squares, namely $3x^2/4+(3x/2+\varepsilon)^2+1$. So the computed difference has the same sign as $\varepsilon$, the function is strictly increasing.
