Suppose a fixed point has coordinates with

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

Problem

Suppose a fixed point

P

has coordinates

(a,b)

with

b>0

, and

Q(x,y)

is a variable point restricted to lie on the

x

-axis. The Euclidean distance from

Q

P

D(x)=\sqrt{(x-a)^2+b^2}.

Answer the following questions about using

D(x)

versus

D^2(x)

when seeking the location of

Q

that is closest to

P

. (a) Write an explicit expression for

D^2(x)

and compute

\dfrac{d}{dx}\bigl[D^2(x)\bigr]

. Do not simplify the derivative. (b) Show algebraically that

\dfrac{d}{dx}\bigl[D(x)\bigr]=0

and

\dfrac{d}{dx}\bigl[D^2(x)\bigr]=0

have exactly the same real solutions. (c) Explain, without further calculation, why minimizing

D^2(x)

is sufficient for finding the value of

x

that minimizes

D(x)

whenever

D(x)>0

. (d) Describe one practical advantage of working with

D^2(x)

instead of

D(x)

in optimization problems of this type.