The curve below is the graph of the function $f$:
Considering this graph, which one of the Taylor series below is the consistent one?
a)$-2-3(x+2)-8(x+2)^2+...$
b)$-2+(x-1)-3(x-1)^2+...$
c)$-2-6(x-1)-5(x-1)^2+...$
d)$-4+(x+1)-8(x+1)^2+...$
e)$x+3x^2+...$
f)$-2+3(x+2)+8(x+2)^2+...$
For me, the correct one is the Taylor series in letter c;
What I've done is to check the first and second derivatives, which is going to give me information about the tangent line and concavity in the follow point a. Since a Taylor series is defined as: $$ \sum_{n=o}^{\infty}\frac{f^{n}(a)}{n!}(x-a)^n $$ I analyzed the first and second derivatives in point a, and the only equation that is making sense is: $ -2-6(x-1)-5(x-1)^2+...$
Can someone please check if the way that I'm doing is indeed correct? If so, the correct answer for this question is c?
Thanks"