A family of horizontal lines (L_k:

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

A family of horizontal lines

L_k:\, y=k

with

0<k<4

intersects the parabola

P:\, y=4-x^{2}

. Let

R_k

denote the closed region that lies (1) below the parabola, (2) above the line

y=k

, and (3) in the first quadrant. Throughout this problem, treat

k

as a variable parameter. (a) Show that the line and the parabola intersect at the two points

(\pm\sqrt{4-k},k)

. State explicitly the positive intersection abscissa

x_{1}(k)

. (b) Write the area

A(k)

R_k

as a definite integral that depends only on

k

, and evaluate the integral to obtain a simplified closed-form expression for

A(k)

. (c) Explain why any value of

k

that produces an extremum of the area must satisfy

A'(k)=0

. Determine all critical values of

k

in the open interval

(0,4)

. (d) Decide whether the absolute maximum of

A(k)

[0,4]

occurs at an endpoint or at a critical value found in part (c). Provide a complete analytic justification.