Suppose we use the least squares criterion to fit a linear model for the following dataset: $(x_1,y_1),...,(x_m,y_m)\in R \times R$, by solving the following optimisation problem:

$$(a^*,b^*) = \text{argmin}_{a,b}\sum^m_{i=1}(y_i-ax_i+b)^2$$

Assume that the solution is unique. Now, my question is, would any of the following statements be true?

i) $\sum^m_{i=1}(y_i-a^*x_i+b^*)y_i = 0$

ii) $\sum^m_{i=1}(y_i-a^*x_i+b^*)x_i^2 = 0$

iii) $\sum^m_{i=1}(y_i-a^*x_i+b^*)x_i = 0$

iv) $\sum^m_{i=1}(y_i-a^*x_i+b^*)^2 = 0$

By my understanding, we have to take the derivative of the loss function wrt $a$ and $b$:

$$\frac{\partial }{\partial a}\sum_{i=1}^m(y_i-ax_i+b)^2 = 0 \\ \Leftrightarrow \frac{\partial }{\partial a}(y_i-a+b) = 0 \\ \Leftrightarrow (y_i-a^*x_i+b^*)x_i = 0$$

So this means that statement iii should be true.

And for $b$:

$$\frac{\partial }{\partial b}\sum_{i=1}^m(y_i-ax_i+b)^2 = 0 \\ \Leftrightarrow (y_i-a^*x_i+b^*) = 0$$

So this means that every statement is true.

Surely my second deduction is incorrect? What am I doing wrong here?