Question5. Исследовать функцию на локальный экстремум (20 баллов):
Studdy Solution
STEP 1
What is this asking? We need to find the *x* and *y* values that make the expression *z* as big or as small as possible! Watch out! Don't forget to check if the points we find are maximums or minimums!
STEP 2
1. Find the partial derivatives
2. Set the partial derivatives to zero
3. Solve the system of equations
4. Check if it's a maximum or minimum
STEP 3
Let's **kick things off** by finding the partial derivative of *z* with respect to *x*.
Imagine *y* is just a constant number, and differentiate with respect to *x*.
We get:
STEP 4
Now, let's find the partial derivative of *z* with respect to *y*.
This time, treat *x* like a constant.
We get:
STEP 5
To find potential maximums or minimums, we set both partial derivatives equal to **zero**:
STEP 6
Now for the **detective work**!
Let's solve this system of equations.
From the first equation, we can express *y* in terms of *x*:
STEP 7
**Substitute** this value of *y* into the second equation:
STEP 8
Now, **isolate** *x*:
STEP 9
**Great!** Now we know .
Let's **plug** this back into the equation we found for *y*:
So, .
STEP 10
To determine if this point is a maximum or minimum, we need to find the second partial derivatives.
Let's **do it**:
STEP 11
Now, let's **calculate** the determinant of the Hessian matrix:
STEP 12
Since and , we have a **local maximum**!
STEP 13
The function *z* has a local maximum at and .
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