Suppose $f: [0,1] \times [0,1] \to \mathbb{R}$ is a differentiable function. Define $g(t) = f(0,t)$ and suppose that $g$ has a maximum in $t_0 \in (0,1)$, and suppose additionally that $D_1f(0,t_0) < 0$. I need to either prove, or give a counter-example to the fact that then $(0,t_0)$ is a local maximum of $f$.
Firstly it follows that $g'(t_0) = D_2f(0,t_0) = 0$. I have set up the Taylor expansion, write $\vec{a} = (0,t_0)$, then: $$f(\vec{a} + \vec{h}) = f(\vec{a}) + D_1f(\vec{a})h_1 + D_2(\vec{a})h_2 + o(\|\vec{h}\|) = f(\vec{a}) + D_1f(\vec{a})h_1 + o(\|\vec{h}\|). $$Now if $h_1 < 0$ it seems that $f(\vec{a} + \vec{h}) > f(\vec{a})$, or at least has the potential to be greater, so my guess is that it does not hold and we need to specify a counter-example. I have tried several but everything fails.
