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Does the function f (x, y) = x³ + y³ + 3xy + 5 have any local maxima or minima? If so, find them.

Does the function f (x, y) = x³ + y³ + 3xy + 5 have any local maxima or minima? If so, find them.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Question:** Does the function \( f(x, y) = x^3 + y^3 + 3xy + 5 \) have any local maxima or minima? If so, find them. **Explanation:** The problem requires us to determine if the given function has any local maxima or minima points. To solve this, we need to find the critical points by setting the partial derivatives of the function equal to zero. After finding the critical points, the second derivative test can be applied to classify them as local maxima, local minima, or saddle points.
Transcribed Image Text:**Question:** Does the function \( f(x, y) = x^3 + y^3 + 3xy + 5 \) have any local maxima or minima? If so, find them. **Explanation:** The problem requires us to determine if the given function has any local maxima or minima points. To solve this, we need to find the critical points by setting the partial derivatives of the function equal to zero. After finding the critical points, the second derivative test can be applied to classify them as local maxima, local minima, or saddle points.
Expert Solution
Step 1

Given that fx,y=x3+y3+3xy+5

The objective is to find the local maxima and minimum.

Let's find fx,fy.

fx=ddxx3+y3+3xy+5=3x2+3y

And to find fy

fy=ddyx3+y3+3xy+5=3y2+3x

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