Show that the function $f(x)=x^3+x+8$ has no relative extrema on $(-,).$

The function has no relative extrema because $f(x)=x^3+x^2+8$ does not equal to $0$ at any point.

The derivation of the function $f(x)=x^3+x+8$ is $f^{\text{'}}(x)=3x^2+1.$ At every point the derivative exists and does not equal to $0.$ So by definition the function has no relative extrema on this interval.

The function has no relative extrema by definition of the derivation.