Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 2

Theorem
If the function $\map h x$ satisfies the equation


 * $\displaystyle -\frac \d {\d x}\sqbrk{\paren{tP+\paren{1-t} }h'}+ tQh=0$

and the boundary conditions


 * $\map h {a,t}=\map h {b,t}=0$

then


 * $\displaystyle\int_a^b\sqbrk{\paren{Ph'^2+Qh^2}t+\paren{1-t} h'^2}\rd x=0$