Let be a real-valued function continuous on the closed interval and differentiable on .

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

Problem

Let

f

be a real-valued function continuous on the closed interval

[a,b]

and differentiable on

(a,b)

. (a) Define a critical point of a real-valued function. (b) Give a clear reason why, when determining the absolute maximum or minimum value of

f

[a,b]

, it is necessary to compare the values of

f

at the endpoints

a, b

with its values at all critical points in

(a,b)

. (c) A student asserts that if

f'(c)=0

for some

c\in(a,b)

, then

f

must attain its absolute maximum at

c

. Construct an explicit example of a function on

[0,3]

that invalidates this assertion, and justify your answer.