We prove the following result: Let X be a real Hilbert space and let J: X → ℝ be a C¹ functional with a nonexpansive derivative. Then, for each r > 0, the following alternative holds: either J’ has a fixed point with norm less than r, or $su{p}_{\left|\right|x\left|\right|=r}J\left(x\right)=su{p}_{{\left|\right|u\left|\right|}_{L²\left(\right[0,1],X)}=r}{\int }_0^{1}J\left(u\left(t\right)\right)dt$.