Chapter 1.2, Problem 14E

The $n$th Taylor polynomial for a function $f$ at $x_0$ is sometimes referred to as the polynomial of degree at most $n$ that "best" approximates $f$ near $x_0$.

1. Explain why this description is accurate.
2. Find the quadratic polynomial that best approximates a function $f$ near $x_0=1$ if the tangent line at $x_0=1$ has equation $y=4x-1,$ and if $f^{\prime \prime }(1)=6.$