Find the Maximum Rate of Change Of $f ( x , y ) = x y ^ { 2 } + \sqrt { y }$

question 30

Multiple Choice

Find the maximum rate of change of $f ( x , y ) = x y ^ { 2 } + \sqrt { y }$ at the point $( 2,1 )$ . In what direction does it occur?

• A) \(\frac { 85 } { 2 } , \left( 2 , \frac { 7 } { 2 } \right)\)
• B) \(\frac { 29 } { 2 } , \left( 1 , \frac { 9 } { 2 } \right)\)
• C) \(\frac { \sqrt { 29 } } { 2 } , \left( 1 , \frac { 5 } { 2 } \right)\)
• D) \(\frac { \sqrt { 85 } } { 2 } , \left( 1 , \frac { 9 } { 2 } \right)\)
• E) none of these

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Definitions:

Unintended Consequences

Outcomes of a policy or action that were not foreseen or intended, often contrary to the original objectives.

Allocating Resources

The process of distributing available resources among various competing needs or projects in an efficient manner.

Invisible Hand

Adam Smith's metaphor describing the self-regulating nature of the marketplace that guides individuals to unintentionally benefit society through personal gain.

Principal-Agent Problem

A dilemma in relationships where one party (the agent) is supposed to act in the best interest of another (the principal) but may have an incentive to act in their own self-interest instead.