Let $f ( x , y ) = a x ^ { 2 } - 2 a x y + 4 y ^ { 2 } - b x$

question 5

Short Answer

Let $f ( x , y ) = a x ^ { 2 } - 2 a x y + 4 y ^ { 2 } - b x$ , where a, b are any positive numbers with a $\gt$ 4.
Find the minimum value of a such that the critical point will be a saddle point.

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Definitions:

Gas Exchange

The biological process through which organisms exchange gases (such as oxygen and carbon dioxide) with their environment, crucial for respiration.

Diffuses

The process by which molecules move from an area of higher concentration to an area of lower concentration.

Viscous

A physical property describing a fluid's resistance to flow, with higher viscosity indicating a thicker and slower-moving fluid.

Dermal Gills

Respiratory structures in some aquatic animals that allow for gas exchange directly through the skin.