Differentiate Min Function I have no clue how to work with min functions, and am struggling! I have a question that has the constraint of Total Income $=150=I_x+I_y$, thus getting $I_y= 150-I_x$. My function is $W = min (U_x, U_y)$. My functions are $U_x = 50 \sqrt(I_x)$ and $U_y = 100\sqrt(I_y)$ Subbing in the constraint you get $W = min(50 \sqrt(I_x), 100\sqrt(150-I_x))$. How do I solve for $I_x$ from here by differentiating and let $W=0$? In a previous question, once subbing in the constraint into the welfare function, the lecturer differentiated the function and made it $=0$ in order to find the point where welfare is maximised. Context: This is welfare economics using a Rawlsian social welfare function.

when does $\sqrt{I_x} \leq 2 \sqrt{150-I_x}$?

By squaring both sides, $I_x \leq 4 (150-I_x)$

$$5I_x \leq 600$$

$$I_x \leq 120$$

$$\min\left(50 \sqrt{I_x}, 100 \sqrt{150-I_X} \right)= \begin{cases} 50 \sqrt{I_X} & I_X \leq 120 \\\ 100 \sqrt{50-I_X}\ & I_X > 120 \end{cases}$$

We can see that the function increases from $0$ to $120$ and then decreases from $120$ onwards.

Hence the maximum point is at $I_X=120$.