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

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

Problem

A function

h

is continuous on the closed interval

[a,b]

and twice differentiable on the open interval

(a,b)

. The function possesses exactly one critical number

c

with

a<c<b

, and it is given that

h'(c)=0

and

h''(c)>0.

(a) Explain, using the geometric interpretations of the first and second derivatives, why

h

must attain a local minimum at

x=c

. (b) Under what additional condition(s) on

h

can you conclude that the value of

h

x=c

is also the absolute minimum value of

h

[a,b]

? Justify your response.