Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a continuos concave function which is continuously differentiable in the first variable. Is it true that
$$ f(x', y') - f(x, y) - f_x(x', y')(x' - x) \geq f(x', y') - f(x', y) $$ $\forall x, x', y, y' \in \mathbb{R}$?
A counterexample is $f(x,y) = -(x-y)^2$ and $(x',y')=(0,0)$. The statement boils down to $(x-y)^2 \geq y^2$ which does not hold for $(x,y) = (1,1)$.