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.