I have the following function to maximize: $$f=\ln(1+(y-x))+\ln(1+(z-x))+\ln(1+(w-x))+\ln(1+(z-y))+\ln(1+(w-y))+\ln(1+(w-z))$$ subject to $x\ge 1, y\ge x, z \ge y, w \ge z, w \le 7$.
WolframAlpha says that local maxima is at $(x_0,y_0,z_0,w_0)=(1, 2.4, 5.6, 7)$.
But $\frac{\partial}{\partial x}(x_0,y_0,z_0,w_0) \approx 0.73 \ne 0$. Why is that?
Also, how do I solve it analytically? Lagrange multipliers? How to handle inequalities?
Thank you.