Let (C ) be the curve given by (y=x^ 3 -3x )

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

Problem

Let

C

be the curve given by

y=x^{3}-3x

. For each real number

k

with

|k|<2

, the horizontal line

y=k

intersects

C

at three distinct points, enclosing a lens-shaped region

S_k

between the curve and the line. Let

A(k)

denote the area of

S_k

. (a) Show that the intersection points satisfy the cubic equation

x^{3}-3x-k=0

, and explain why the condition

|k|<2

guarantees exactly three real roots. (b) Let

a(k)<b(k)

be the leftmost and rightmost intersection abscissae. Express

A(k)

as a single definite integral in terms of

a(k)

b(k)

, and

k

. (c) Differentiate

A(k)

with respect to

k

to obtain

A'(k)

. Find all critical values of

k

(-2,2)

. (d) Use a second-derivative test or other analytic argument to classify each critical value found in part (c) and determine the value of

k

that yields the maximum area.