Let $f$ be a function with $f(0)=1$, $f'(0)=2$, and $f''(0)=-2$. Which of the following could be the graph of the second-degree Taylor polynomial for $f$ about $x=0$ ?

Summary Submit

The graph is concave down, increasing, and has a value of $1$ at $x=0$.

Since there are two graphs that match this description, we can use the equation to find a few other points.

The equation of the Taylor polynomial is:

$$P(x)= 1\left(\frac{x^0}{0!}\right) + 2\left(\frac{x^1}{1!}\right)-2\left(\frac{x^2}{2!}\right)$$ $$P(x)= 1+2x-x^2$$

Choosing an arbitray point for testing:

$$P(1)=2$$

This point is only valid for one of the graphs.