Solution: Find the absolute maximum and minimum values of

Chapter 13, Problem 64E

(choose chapter or problem)

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

a. critical points (points where \(d f / d t\) 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.

Functions:

a. \(f(x, y)=x^{2}+y^{2}\)

b. \(g(x, y)=1 /\left(x^{2}+y^{2}\right)\)

Curves:

i) The line \(x=t, y=2-2 t\)

ii) The line segment \(x=t, y=2-2 t, 0 \leq t \leq 1\)

Equation Transcription:

Text Transcription:

f(x,y)

x=x(t),y=y(t)

f

t

df/dt

f

df/dt

f(x,y)=x^2+y^2

g(x,y)=1/x^2+y^2

x=t, y=2-2t

x=t, y=2-2t, 0 leq t leq 1

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