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

question 107

Multiple Choice

Determine whether F is conservative. If so, find a function f such that $\mathbf { F } = \nabla f$ . $\mathbf { F } ( x , y ) = \left( 4 x ^ { 3 } + 2 y \right) \mathbf { i } + \left( 3 x - 4 y ^ { 3 } \right) \mathbf { j }$

• A) \(f ( x , y ) = x ^ { 4 } + 2 y ^ { 2 } + 3 x ^ { 2 } - y ^ { 4 } + C\)
• B) not conservative
• C) \(f ( x , y ) = x ^ { 4 } + y ^ { 4 } + C\)
• D) \(f ( x , y ) = 4 x ^ { 3 } + 2 y + 3 x - 4 y ^ { 3 } + C\)

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Definitions:

T Test

A statistical test used to compare the means of two groups to determine if they are significantly different from each other.

P < 0.05

Indicates that the probability of obtaining the observed data, or more extreme, under the null hypothesis is less than 5%, suggesting statistically significant results.

Significant Correlation

A statistical relationship between two variables that is unlikely to be due to chance at a specified level of significance.

Causality

The relation between a cause and its effect, often established through a combination of observational studies and experiments.