Aus Online Mathematik Brückenkurs 1

If we complete the square

\displaystyle x^{2}+x+1=\left( x+\frac{1}{2} \right)^{2}-\left( \frac{1}{2} \right)^{2}+1=\left( x+\frac{1}{2} \right)^{2}+\frac{3}{4}

we see on the right-hand side that we can make the expression arbitrarily large simply by choosing \displaystyle x+\frac{1}{2} sufficiently large. Hence, there is no maximum value.