Chapter 4.8, Problem 1P

Use the Taylor series about $x=a$ to verify the familiar "second derivative test" for a maximum or minimum point. That is, show that if $f^{\prime }(a)=0,$ then $f^{\prime \prime }(a)>0$ implies a minimum point at $x=a$ and $f^{\prime \prime }(a)<0$ implies a maximum point at $x=a$ Hint: For a minimum point, say, you must show that $f(x)>f(a)$ for all $x$ near enough to $a$.