Prove that the profits of the firm weakly decreases with input prices
From FOC, we know that:
\begin{align}
\nabla_x\pi(\mathbf{x},\mathbf{w})=p\nabla f(\mathbf{x})-\mathbf{w}=\mathbf{0} \tag{1}
\end{align}
This will be true at equilibrium, i.e. for any given $\mathbf{w}$, the input vector $\mathbf{x}$ will adjust so that the above holds.
Now consider $d\pi(\mathbf{x},\mathbf{w})/d w_i$ (and using $(1)$):
\begin{align}
\frac{d\pi(\mathbf{x},\mathbf{w})}{d w_i} &amp;=\nabla_x\pi\cdot\nabla_{w_i} \mathbf{x} \, + \nabla_w\pi\cdot\nabla_{w_i} \mathbf{w} \\
&amp;=0 +\nabla_w\pi\cdot\nabla_{w_i} \mathbf{w} \tag{*}\\
&amp;=\nabla_w(pf(\mathbf{x})-\mathbf{w}\cdot \mathbf{x)} \cdot \nabla_{w_i} \mathbf{w} \\
&amp;= \Big(p\nabla_wf(\mathbf{x})-\nabla_w(\mathbf{w}\cdot \mathbf{x)}\Big)\cdot \nabla_{w_i} \mathbf{w} \\
&amp;= \Big(p \frac{\partial\mathbf{x}}{\partial\mathbf{w}}\nabla f(\mathbf{x})-\frac{\partial\mathbf{x}}{\partial\mathbf{w}}\mathbf{w} - \mathbf{x}\Big)\cdot \nabla_{w_i} \mathbf{w}\\
&amp;= \mathbf{J}(p\nabla f(\mathbf{x})-\mathbf{w})\cdot \nabla_{w_i} \mathbf{w}  -\mathbf{x} \cdot \nabla_{w_i} \mathbf{w} \\
&amp;= \mathbf{J} \nabla_x\pi(\mathbf{x},\mathbf{w})\cdot \nabla_{w_i} \mathbf{w}-\mathbf{x} \cdot \nabla_{w_i} \mathbf{w} \tag{**}\\
&amp;= -\mathbf{x} \cdot \nabla_{w_i} \mathbf{w} \\
&amp;= -x_i 
\end{align}
where, in steps $(*), (**)$, FOC is used and $\mathbf{J}$ is the Jacobian matrix.
Since, $x_i \geq0$, we have:
$$\frac{d\pi(\mathbf{x},\mathbf{w})}{d w_i} \leq0$$