# Real analysis – \$ u \$ can not reach the maximum if it satisfies a linear differential equation \$ Lu geq 0 \$

I have a function $$u in C ^ 2 ((a, b)) ​​cap C ^ 0 ([a,b])$$ and a limited function $$g: (a, b) rightarrow mathbb {R}$$, whereas $$(a, b) subset mathbb {R}$$

Now consider the linear operator $$L: = frac {d ^ 2} {dx ^ 2} + g frac {d} {dx}$$ , I would like to show that, though $$Lu geq 0$$ then either $$u$$ is constant or can not reach its maximum $$(a, b)$$,

I can show that though $$Lu> 0$$ then $$u$$ can not reach its maximum
because if there is a maximum in $$(a, b)$$ let's say $$x_0$$, then $$Lu (x_0) = u & # 39; & gt; (x_0) + g (x_0) u & # 39; (x_0) leq 0$$ (using the fact that the second derivative must be at most negative and that the first derivative must be zero)

I am not sure to prove the case $$Lu geq 0$$ and I'm also pretty unsure where the limits are $$g$$ would be important to have. Hints are much appreciated.