Find a Function $f ( x , y )$ Such That $\nabla f = \left( x ^ { 2 } + y \right) \mathbf { i } + ( x + y ) \mathbf { j }$

question 186

Multiple Choice

Find a function $f ( x , y )$ such that $\nabla f = \left( x ^ { 2 } + y \right) \mathbf { i } + ( x + y ) \mathbf { j }$ .

• A) \(\frac { 1 } { 3 } x ^ { 3 } + x y + y ^ { 2 }\)
• B) \(\frac { 1 } { 3 } x ^ { 3 } + \frac { 1 } { 2 } x y + \frac { 1 } { 2 } y ^ { 2 }\)
• C) \(\frac { 1 } { 3 } x ^ { 3 } + x y + \frac { 1 } { 2 } y ^ { 2 }\)
• D)\(\frac { 1 } { 3 } x ^ { 3 } + \frac { 1 } { 2 } x y + y ^ { 2 }\)
• E) \(\frac { 1 } { 2 } x ^ { 3 } + x y + y ^ { 2 }\)
• F)\(\frac { 1 } { 2 } x ^ { 3 } + \frac { 1 } { 2 } x y + \frac { 1 } { 2 } y ^ { 2 }\)
• G) \(\frac { 1 } { 2 } x ^ { 3 } + x y + \frac { 1 } { 2 } y ^ { 2 }\)
• H) \(\frac { 1 } { 2 } x ^ { 3 } + \frac { 1 } { 2 } x y + y ^ { 2 }\)

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Definitions:

Electroconvulsive Therapy

A psychiatric treatment in which seizures are electrically induced in patients to provide relief from mental disorders, particularly major depressive disorder and schizophrenia.

Severe Depression

A serious mental health condition characterized by persistent feelings of sadness, hopelessness, and a lack of interest or pleasure in activities.

Electroconvulsive Therapy

A psychiatric treatment in which seizures are electrically induced in patients to provide relief from severe mental disorders.

Schizophrenia

A chronic and severe mental disorder that affects how a person thinks, feels, and behaves, leading to a range of psychological symptoms.