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:

Self-efficacy

An individual’s belief in their own ability to successfully accomplish tasks and reach goals.

Spotlight Effect

The idea that one's actions and appearance draw more observation from others than truly happens.

Self-serving Bias

A common cognitive bias that leads individuals to attribute success to personal factors while attributing failures to external factors, protecting self-esteem.

Disparage

To belittle or depreciate someone or something; often reflecting a perceived difference in value or status.