Find the Gradient Vector Field Of $f$ Select the Correct Answer

question 124

Multiple Choice

Find the gradient vector field of $f$ . Select the correct answer.
$$f ( x , y , z ) = \sqrt { x ^ { 2 } + 4 y ^ { 2 } + 6 z ^ { 2 } }$$

• A) \(\frac { y } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { i } + \frac { x } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { j } + \frac { z } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { k }\)
• B) \(\frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { i } + \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { j } + \frac { z } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { k }\)
• C)
• \(\frac { 2 x } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { i } + \frac { 2 y } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { j } + \frac { 2 z } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { k }\)
• D)
• \(\frac { x } { \sqrt { x ^ { 2 } + 4 y ^ { 2 } + 6 z ^ { 2 } } } \mathbf { i } + \frac { 4 y } { \sqrt { x ^ { 2 } + 4 y ^ { 2 } + 6 z ^ { 2 } } } \mathbf { j } + \frac { 6 z } { \sqrt { x ^ { 2 } + 4 y ^ { 2 } + 6 z ^ { 2 } } } \mathbf { k }\)
• E) None of these

Identify factors of production and their roles in the economy.

Comprehend the basic mechanisms of trade and barriers to trade.

Understand the circular-flow model of an economy and its components.

Recognize the significance of specialization and trade in improving economic efficiency.

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Definitions:

Produced Units

The quantity of finished goods a company has manufactured during a specific time period.

Maintenance Cost

Expenses incurred to keep property, plant, and equipment in operating condition, ensuring they can perform their intended functions.

Cost Formula

A mathematical expression or equation used to calculate the total cost of producing goods or services, based on factors like raw material, labor, and overhead costs.

Planned Machine Time

The scheduled period during which machinery is expected to operate in order to meet production targets, integral to manufacturing efficiency and planning.