Consider the curve for and the fixed point

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

Problem

Consider the curve

y=x^{2}

for

x\in\mathbb R

and the fixed point

P(0,3)

. A point

Q(t,t^{2})

moves along the curve. Your goal is to locate the point on the parabola that is closest to

P

. (a) Show that the square of the distance between

P

and

Q

can be written as

s^{2}(t)=t^{2}+(t^{2}-3)^{2}

. (b) Compute

\dfrac{ds^{2}}{dt}

and determine all critical points of

s^{2}(t)

. (c) Use a second-derivative test or another appropriate method to verify that the critical point you select corresponds to a minimum distance. (d) Find the exact coordinates of the point on the parabola that is closest to

P

and state the minimum distance.

Related theory

Optimization
Applied optimization
Maxima and minima

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