Consider the Function $g ( x , y ) = x ^ { \frac { 1 } { 5 } } y ^ { \frac { 1 } { 5 } }$

question 88

Essay

Consider the function $g ( x , y ) = x ^ { \frac { 1 } { 5 } } y ^ { \frac { 1 } { 5 } }$ (a)Find g_x(x, y)and g_y(x, y)for (x, y) $\neq$ (0, 0).
(b)Use the limit definition of partial derivative to show that g_x(0, 0)= 0 and g_y(0, 0)= 0.
(c)Are the functions g_x and g_y continuous at (0, 0)? Explain.
(d)Is g differentiable at (0, 0)? Explain.

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Definitions:

Share Capital

The total value of a company's shares of stock that have been authorized, issued, and purchased by investors.

Financial Statements

Formal records that outline the financial activities and conditions of a business, entity, or individual, including the balance sheet, income statement, and cash flow statement.

Retained Earnings

Funds that are not distributed as dividends but are reserved by the company to reinvest in its business, pay off debt, or save for future use.