Find All the Local Maxima, Local Minima, and Saddle Points $$f ( x , y ) = 6 x ^ { 2 } y + 5 x y ^ { 2 }$$

question 8

Multiple Choice

Find all the local maxima, local minima, and saddle points of the function.
- $$f ( x , y ) = 6 x ^ { 2 } y + 5 x y ^ { 2 }$$

• A) \(f ( 30,30 ) = 297,000\), local maximum
• B) \(\left\{ \left( \frac { 1 } { 5 } , \frac { 1 } { 6 } \right) = \frac { 61 } { 900 } \right.\), local minimum
• C) \(\mathrm { f } ( 0,0 ) = 0\), saddle point
• D) \(\left\{ \left( \frac { 1 } { 6 } , \frac { 1 } { 5 } \right) = \frac { 1 } { 15 } \right.\), local minimum

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Definitions:

Long-Run Supply

A period sufficient for all inputs to production, including capital, to be varied, allowing for adjustments to changes in demand or technology.

Average Total Cost

The total cost of production divided by the total quantity produced, representing the per-unit cost of production.

Long-Run Equilibrium

A state in which all factors of production and economic variables are in balance, with no external pressures forcing change.

Demand Decreases

A situation where the quantity of a product or service that consumers are willing to buy at a given price drops, often due to changes in preferences, income, or substitutes.