Answer: Theory and Examples.Extreme Values on Parametrized

Chapter 14, Problem 63E

(choose chapter or problem)

Problem 63E

Theory and Examples.

Extreme Values on Parametrized Curves To find the extreme values of a function ƒ(x, y) on a curve x = x(t), y = y(t), we treat ƒ as a function of the single variable t and use the Chain Rule to find where df/dt is zero. As in any other single-variable case, the extreme values of ƒ are then found among the values at the

a. critical points (points where dƒ/ dt is zero or fails to exist), and

b. endpoints of the parameter domain.

Find the absolute maximum and minimum values of the following functions on the given curves.

Function: ƒ(x, y) = xy

Curves:

i) The line x = 2t, y = t + 1

ii) The line segment

iii) The line segment