Science:Math Exam Resources/Courses/MATH100 B/December 2023/Question 25/Solution 1

The two residual values we have for the model ${\textstyle y(x)=ax}$ are ${\textstyle y(2)-2}$ and ${\textstyle y(3)-2}$. The sum of squared residuals is

$$R=(y(2)-2)^{2}+(y(3)-2)^{2}=(2a-2)^{2}+(3a-2)^{2}.$$

which is quadratic in ${\textstyle a}$ and concave up, so we know that the zero of the derivative of ${\textstyle R}$ with respect to ${\textstyle a}$ will correspond to a minimum. The derivative is

$${\begin{aligned}{\frac {d}{da}}R&=2(2a-2)\cdot 2+2(3a-2)\cdot 3\\&=8a-8+18a-12\\&=26a-20.\end{aligned}}$$

Setting the derivative to zero and solving gives ${\textstyle a=20/26=10/13}$.