A function g is continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Suppose that g′(c)=0 at a single point c with a<c<b and that g′′(c)>0. No other critical points exist in (a,b).
(a) State the relationship between the sign of g′′(c) and the nature of the critical point at x=c. Provide a brief justification based on the behavior of g′ near c.
(b) Explain why the absence of any additional critical points in (a,b) forces g(x) to be strictly decreasing on one side of c and strictly increasing on the other side.
(c) Combine your results from parts (a) and (b) to justify that g attains its absolute minimum at x=c on [a,b] without comparing numerical function values.