Find the Gradient Field of the Function $f ( x , y , z ) = e ^ { x ^ { 6 } + y ^ { 5 } + z ^ { 7 } }$

question 129

Multiple Choice

Find the gradient field of the function.
- $f ( x , y , z ) = e ^ { x ^ { 6 } + y ^ { 5 } + z ^ { 7 } }$

• A) \(\nabla f = x ^ { 6 } e ^ { x ^ { 6 } + y ^ { 5 } + z ^ { 7 } } i + y ^ { 5 } e ^ { x ^ { 6 } + y ^ { 5 } + z ^ { 7 } } j + z ^ { 7 } e ^ { x ^ { 6 } + y ^ { 5 } + z ^ { 7 } } \mathbf { k }\)
• B) \({\nabla} f = 6 \mathrm { x } ^ { 5 } \mathrm { e } ^ { \mathrm { x } ^ { 6 } } \mathbf { i } + 5 \mathrm { y } ^ { 4 } \mathrm { e } ^ { 5 } \mathbf { j } + 7 \mathrm { z } ^ { 6 } \mathrm { e } ^ { \mathrm { z } ^ { 7 } } \mathbf { k }\)
• C) \({\nabla} f = 6 x ^ { 5 } e ^ { x ^ { 6 } } + y ^ { 5 } + z ^ { 7 } i + 5 y ^ { 4 } e ^ { x ^ { 6 } } + y ^ { 5 } + z ^ { 7 } j + 7 z ^ { 6 } e ^ { x ^ { 6 } } + y ^ { 5 } + z ^ { 7 } k\)
• D) \(\nabla f = x ^ { 5 } e ^ { x ^ { 6 } + y ^ { 5 } + z ^ { 7 } } i + y ^ { 4 } e ^ { x ^ { 6 } + y ^ { 5 } + z ^ { 7 } } j + z ^ { 6 } e ^ { x ^ { 6 } + y ^ { 5 } + z ^ { 7 } } \mathbf { k }\)

---

Definitions:

Barriers To Entry

Economic or legislative obstacles that prevent new competitors from easily entering an industry or area of business.

Monopoly Power

The degree of power held by a singular entity to control prices or exclude competition within a market.

Natural Monopoly

A market condition where a single firm can supply a good or service to an entire market more efficiently than multiple firms due to high fixed or startup costs.

Economies Of Scale

The cost advantage that arises with increased output of a product, as fixed costs are spread over more units of production, reducing the cost per unit.