Question on Lagrange multipliers , don't understand the solutions. An error in the solution maybe?

Question

So I've been trying to solve the question above. The thing I don't get is that why is F(t,u,λ) = h(t,u) + λg(t,u)? Shouldn't it be -λg(t,u) instead?

Answer 1

Here $\lambda$ is an unknown. If you prefer, you can switch to $\bar{\lambda}=-\lambda$ and get $F(t,u,\bar{\lambda})=h(t,u)-\bar{\lambda}g(t,u)$. If you find $\bar{\lambda}$, you find also $-\bar{\lambda}=\lambda$.

From a geometric viewpoint, what you request is that the gradient of $h$ be parallel to the gradient of the constraint $g$: no matter if they point in the same direction.