a) A tangent line can be used as a linear approximation to a function near

a point, but it is not the only such approximation, so the answer is no -

a linear approximation to a function is not necessarily a tangent line.

b) At least one of the definitions of a point of inflection for a function f is

that it is a point at which the f'' changes sign so of necessity f''(x)=0 at

a point of inflection, but it is not a sufficient condition. As Glaysher points

out every point on f(x)=mx+c is a point where f''(x)=0, but they are not

points of inflection. So f''(a)=0, is not sufficient to guarantee that x=a is

a point of inflection of f.

RonL