Question about using the gradient vector to bound a function.

Question

Say we have a function $v(x,z)$ with continuous partial derivatives, where the domains of $x$ and $z$ are the non-negative reals, and $v(0,0)=0$. We also have another differentiable function $g(z)$ with $g(0)=0$. We want to prove that $v(g(z),z)\geq v(z,z)$. Since the linear approximation to these values can be computed from $v(0,0)=0$ along the vectors $(g'(z),1)$ and $(1,1)$ respectively, given that the gradient vector $(\frac{\partial v}{\partial x},\frac{\partial v}{\partial z})$ gives the path of steepest increase in $v$, is it sufficient to show that $\frac{\partial v}{\partial x} \leq \frac{\partial v}{\partial z}$ to prove the result?

Answer 1

What about: $q_1(z) = v(g(z),z) \geq v(z,z) = q_2(z)$.

We know: $q_1(0) \geq q_2(0)$. So, we just want to show $\partial_zq_1 \geq \partial_zq_2$.

But then: $\partial_zq_1=\partial_x v\partial_z g + \partial_zv\geq \partial_x v + \partial_z v = \partial_zq_2$.

Which is equivalent to: $ \partial_x v\partial_z g \geq \partial_x v $.