Past exam of the partial differential equations for pure and applied mathematicians and calculus of variations course of the University of Oxford / 2008 / B / 8 / a / i / Solution

The continuous superlinear function $\Phi$ tends to positive infinity at both ends, so it has a finite global minimum $M$. Hence $I(u)\ge M(b-a)$ for every admissible function. An affine function with the prescribed endpoints is Lipschitz and has finite energy by continuity of the integrand on its compact range. Thuswhere the strict left inequality simply records finiteness of $M$.