Mathematics

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.

Intuitive explanation

This problem asks you to recognize a powerful optimization shortcut: minimizing distance is equivalent to minimizing squared distance, since the square root function is monotonically increasing. By working with $D^2(x) = (x-a)^2 + b^2$ instead of $D(x)$ , you avoid messy calculus with square roots while finding the exact same minimizer—the key insight is that removing a monotonic wrapper doesn't change where the minimum occurs. This is a practical lesson in choosing the easiest path to the same answer.

Intuitive explanation

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