Find All the Second Order Partial Derivatives of the Given $$\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = x \mathrm { y } ^ { 2 } + y \mathrm { e } ^ { \mathrm { x } ^ { 2 } + 5 }$$

question 74

Multiple Choice

Find all the second order partial derivatives of the given function.
- $$\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = x \mathrm { y } ^ { 2 } + y \mathrm { e } ^ { \mathrm { x } ^ { 2 } + 5 }$$

• A) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 2 y \mathrm { e } ^ { \mathrm { x } ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = 2 \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = 2 \mathrm { y } + 2 x e ^ { \mathrm { x } ^ { 2 } }\)
• B) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 2 y \mathrm { e } ^ { x ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial y ^ { 2 } } = 2 \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = 2 x \mathrm { e } ^ { \mathrm { x } ^ { 2 } }\)
• C) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = y \mathrm { e } ^ { \mathrm { x } ^ { 2 } } \left( 1 + 2 \mathrm { x } ^ { 2 } \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \mathrm { y } + x \mathrm { x } ^ { 2 }\)
• D) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 2 \mathrm { ye } ^ { \mathrm { x } ^ { 2 } } \left( 1 + 2 \mathrm { x } ^ { 2 } \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = 2 \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = 2 \mathrm { y } + 2 \mathrm { xe } ^ { \mathrm { x } ^ { 2 } }\)

Understand the concepts of payback method and time value of money in capital budgeting decisions.

Identify the role of preference decisions in the context of capital budgeting.

Comprehend that shorter payback periods do not necessarily indicate higher profitability.

Grasp the significance of aligning the discount rate with the company's minimum required rate of return in net present value analysis.

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