Show That At $\left( \frac { 1 } { \sqrt { 2 } } , 0 \right)$

question 193

Essay

Show that at $\left( \frac { 1 } { \sqrt { 2 } } , 0 \right)$ , the equation $x ^ { 2 } + \frac { 1 } { 2 } y ^ { 2 } + \frac { 1 } { 3 } z ^ { 2 } = 1$ defines $Z$ implicitly as a function of $x$ and $y$ , and then compute $\frac { \delta z } { \delta x } \left( \frac { 1 } { \sqrt { 2 } } , 0 , \sqrt { \frac { 3 } { 2 } } \right)$ and $\frac { \delta z } { \delta y } \left( \frac { 1 } { \sqrt { 2 } } , 0 , \sqrt { \frac { 3 } { 2 } } \right)$ .

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Definitions:

Nonlabor Income

Income received from sources other than direct employment or labor, such as investments, retirement pensions, or social security benefits.

Wage Rate

The standard amount of compensation given for the performance of specific services, typically expressed per hour, day, or other periods of work time.

Budget Equation

An equation that represents the relationship between income, savings, and expenses.

Leisure

Time that an individual can spend at their discretion rather than working, often used in economic models to discuss trade-offs between work and free time.