For each positive real number (k )

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

For each positive real number

k

, let

R_k

be the region in the first quadrant that is bounded by the curves

y=x

and

y=kx^{2}

between their points of intersection. The area of this region is denoted by

A(k)

. (a) Find all points of intersection of

y=x

and

y=kx^{2}

when

k>0

. (b) Derive a simplified formula for

A(k)

expressed only in terms of

k

. (c) Determine the value of

k>0

that minimizes

A(k)

. Include all necessary calculus justifications. (d) Briefly interpret how the value of

k

found in part (c) reflects the influence of the quadratic coefficient on the size of the enclosed region.