Which of the following function is the Linearization of the functionf(x,y)=x2−xy+12y2+3
at (3,2)
?

Question 20Answer

a.
x−4y−2

b.
4x−y+2

c.
4x−y−2

d.
x−4y+2

To find the linearization of the function \( f(x, y) = x^2 - xy + \frac{1}{2}y^2 + 3 \) at the point \((3, 2)\), we need to calculate the partial derivatives with respect to \( x \) and \( y \) and then use the formula for the linear approximation.

### Step 1: Compute the partial derivatives

1. **Partial derivative with respect to \( x \):**
   \[
   f_x(x, y) = \frac{\partial}{\partial x}\left( x^2 - xy + \frac{1}{2}y^2 + 3 \right) = 2x - y
   \]

2. **Partial derivative with respect to \( y \):**
   \[
   f_y(x, y) = \frac{\partial}{\partial y}\left( x^2 - xy + \frac{1}{2}y^2 + 3 \right) = -x + y
   \]

### Step 2: Evaluate the partial derivatives at the point \((3, 2)\)

1. \( f_x(3, 2) = 2(3) - 2 = 6 - 2 = 4 \)
2. \( f_y(3, 2) = -(3) + 2 = -3 + 2 = -1 \)

### Step 3: Write the linearization formula

The linearization \( L(x, y) \) of \( f(x, y) \) at the point \((3, 2)\) is given by:
\[
L(x, y) = f(3, 2) + f_x(3, 2)(x - 3) + f_y(3, 2)(y - 2)
\]

### Step 4: Evaluate \( f(3, 2) \)

\[
f(3, 2) = 3^2 - 3(2) + \frac{1}{2}(2)^2 + 3 = 9 - 6 + 2 + 3 = 8
\]

### Step 5: Substitute the values into the linearization formula

\[
L(x, y) = 8 + 4(x - 3) - 1(y - 2)
\]

Simplify the expression:

\[
L(x, y) = 8 + 4x - 12 - y + 2
\]
\[
L(x, y) = 4x - y - 2
\]

### Conclusion:
The linearization of the function at \((3, 2)\) is \( L(x, y) = 4x - y - 2 \).

Therefore, the correct answer is:
**c. \( 4x - y - 2 \)**

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Would you like further details or have any questions? 

Here are 5 related questions you might be interested in:
1. How do you find the linearization of a function at a different point?
2. What is the geometric interpretation of a linearization?
3. How do partial derivatives help in linear approximations?
4. Can linearization be used for functions of more than two variables?
5. What are some practical applications of linearization in engineering?

**Tip:** Linearization is especially useful in approximating the value of a function near a given point.

Learn how to find the linearization of the function f(x, y) = x^2 - xy + 1/2 y^2 + 3 at the point (3, 2). Step-by-step solution involving partial derivatives and linear approximation. Suitable for advanced undergraduate level.

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