Determine Whether $\mathbf { F }$ Is Conservative $f$ Such That $\mathbf { F } = \nabla f$

question 51

Multiple Choice

Determine whether $\mathbf { F }$ is conservative. If so, find a function $f$ such that $\mathbf { F } = \nabla f$ .
Select the correct answer.
$$\mathbf { F } ( x , y ) = \left( 5 x ^ { 4 } + 4 y \right) \mathbf { i } + \left( 7 x - 5 y ^ { 4 } \right) \mathbf { j }$$

• A) \(f ( x , y ) = 5 x ^ { 4 } + 4 y + 7 x - 5 y ^ { 4 } + C\)
• B) not conservative
• C) \(f ( x , y ) = x ^ { 5 } + y ^ { 5 } + C\)
• D) \(f ( x , y ) = x ^ { 5 } + 4 y ^ { 2 } + 7 x ^ { 2 } - y ^ { 5 } + C\) .

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Definitions:

Distinctive Values

Unique or specific beliefs and principles that differentiate one culture, society, or individual from another, guiding behaviors and decisions.

Larger Culture

The dominant set of norms, values, beliefs, and practices that characterizes a larger society or population group.

Rationalization

The process of making actions or outcomes seem reasonable or logical, often by creating a plausible explanation.

Efficient Means

Methods or strategies that achieve desired outcomes with the least amount of waste, effort, or expense.