Question

We have the following function

z=-x³/12-4 y²+6 x y+20 x-2 y
(a) Find the stationary values. That is, solve the first order conditions for the stationary values

Answer 1

**Identification of Elements:**

1. **Start/End:** Oval shape representing the start and end of the flowchart.

2. **Process:** Rectangle indicating a process or computation step.

3. **Input/Output:** Parallelogram representing input or output operations.

4. **Decision:** Diamond shape for decision-making points.

5. **Arrow:** Arrows connecting the elements to show the flow of the process.

**Flowchart:**

1. **Start:** Oval

2. **Input:** Rectangle (Input x, y)

3. **Process:** Rectangle (Calculate z=-x³/12-4y²+6xy+20x-2y)

4. **Output:** Parallelogram (Display z)

5. **Decision:** Diamond (Check if first-order conditions are met)

6. **Process:** Rectangle (If yes, display stationary values; if no, end)

7. **End:** Oval

**Pseudo Code:**

```

Start

Input x, y

z = -x³/12 - 4y² + 6xy + 20x - 2y

Display z

If first-order conditions are met:

Display stationary values

Else:

End

End

```

Final Answer

Stationary values are found by solving the first-order conditions.

Step-by-step explanation

To find stationary values, we need to calculate the first-order partial derivatives of the given function with respect to x and y. Setting these derivatives equal to zero and solving for x and y will provide the stationary points. Let
`\( (\partial z)/(\partial x) = -(x^2)/(4) + 6y + 20 \) and \( (\partial z)/(\partial y) = -8y + 6x - 2 \).`Setting both of these equal to zero and solving simultaneously will yield the values of x and y at the stationary points.

For example, solving
`\( -(x^2)/(4) + 6y + 20 = 0 \) and \( -8y + 6x - 2 = 0 \) s`imultaneously will give the values of x and y. Once these values are found, they can be substituted back into the original function to obtain the corresponding z values at the stationary points.

In summary, the stationary values are obtained by solving the system of equations formed by setting the partial derivatives of the function equal to zero.