Let $f ( x , y , z ) = x ^ { 2 } y + y ^ { 3 } z + x z ^ { 3 }$

question 43

Essay

Let $f ( x , y , z ) = x ^ { 2 } y + y ^ { 3 } z + x z ^ { 3 }$ and let $P ( 2,1 , - 1 )$ .(a) Find the directional derivative at $P$ in the direction of $( 1,2,3)$ .(b) In what direction does $f$ increase most rapidly?
(c) What is the maximum rate of change of $f$ at the point $P$ ?

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Definitions:

Marginal Cost

The cost incurred by producing one additional unit of a product or service.

Output

The total amount of goods or services produced by an individual, company, or economy during a specific period.

Negative Marginal Returns

Occurs when adding an additional factor of production actually decreases the total output, which can happen when there is too much input for the available resources or technology.

Input

Resources used in the production process to create goods or services, including labor, raw materials, and capital.