Sample Solution from
Microeconomic Theory
12th Edition
ISBN: 9781337517942
Chapter 2
Problem 2.1P
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a.

To determine

The partial derivatives of the function f(x,y) .

Expert Solution

Answer

The partial derivatives of the function f(x,y) are

  ∂f∂x=8x,∂f∂y=6y,∂2f∂x2=8,∂2f∂y2=6,∂2f∂x∂y=0

Explanation of Solution

Given information:

The function,

  f(x,y)=4x2+3y2

Since,

  f(x,y)=4x2+3y2

Partially differentiate with respect to x and y ,

  ∂f∂x=∂∂x(4x2+3y2)=4×2x+0=8x

and

  ∂f∂y=∂∂y(4x2+3y2)=0+3×2y=6y

Now

  ∂2f∂x2=∂∂x(∂f∂x)=∂∂x(8x)=8×1=8

and

  ∂2f∂y2=∂∂y(∂f∂y)=∂∂y(6y)=6×1=6

Also,

  ∂2f∂x∂y=∂∂x(∂f∂y)=∂∂x(6y)=0

b.

To determine

The derivative dydx using the implicit function theorem.

Expert Solution

Answer

  dydx=−8x6y

Explanation of Solution

Given information:

The function,

  f(x,y)=4x2+3y2

and

  f(x,y)=16

Since,

  f(x,y)=4x2+3y2

Then,

  16=4x2+3y2

Differentiate with respect to x ,

  0=4×2x+3×2ydydx8x+6ydydx=06ydydx=−8xdydx=−8x6y

c.

To determine

The value of dydx at x=1, y=2 .

Expert Solution

Answer

  dydx=−23

Explanation of Solution

Since,

  dydx=−8x6y

Substituting x=1, y=2 ,

  dydx=−8×16×2

  dydx=−812=−23

d.

To determine

To draw:The graph of dydx=−23 on the surface f(x,y)=4x2+3y2 .

Expert Solution

Explanation of Solution

Since, dydx is the slope of the curve,

From the above part,

  dydx=−83

So, draw the tangent plane on the surface

  f(x,y)=4x2+3y2

  

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