Cant understand how we got this equation

Question

I was going through a tutorial that introduces cubic splines. A snapshot of the tutorial is as follows :

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Now I dont understand how we got:

$y_1''=6a_1(x_1-x_1)+2b_1=0 + 2b_1=2b1$

I dont understand how

$6a_1(x_1-x_1)+2b_1=0 + 2b_1$ or how $6a_1(x_1-x_1)=0$

Any help on how we got this would be appreciated:

Answer 1

On the segment where $x_1 \le x \le x_2$, we have $$ y''(x) = 6a_1(x-x_1) + 2b_1 $$ We want to impose the constraint that $y''(x_1)$ is equal to some given number $y''_1$. Then $$ y''_1 = y''(x_1) \Longrightarrow y''_1 = 6a_1(x_1-x_1) + 2b_1 $$ But $6a_1(x_1-x_1) = 6a_1 \times 0 = 0$, so we get $$ y''_1 = 0 + 2b_1 = 2b_1 $$