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.