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.