A function is continuous on the closed interval and differentiable on the open interval

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

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

[a,b]

without comparing numerical function values.