Find the critical points of the function f(x,y)=y^3 -x^3 -2xy+5. For each critical point of f, determine whether it is a local minimum, local maximum, or a saddle point.

daniel suriya

daniel suriya

Answered question


Answer & Explanation

Jeffrey Jordon

Jeffrey Jordon

Expert2022-11-07Added 2605 answers

Move all the expressions to the left side of the equation.

Subtract y3 from both sides of the equation.


Add x3 to both sides of the equation.


Add 2xy to both sides of the equation.


Subtract 5 from both sides of the equation.


Find the first derivative.

By the Sum Rule, the derivative of f(x,y)-y3+x3+2xy-5 with respect to f is ddf[f(x,y)]+ddf[-y3]+ddf[x3]+ddf[2xy]+ddf[-5].


Evaluate ddf[f(x,y)].


Differentiate using the Constant Rule.


Combine terms.


The first derivative of f(x) with respect to x is (x,y).


Set the first derivative equal to 0.


Find the values where the derivative is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

There are no values of x in the domain of the original problem where the derivative is 0 or undefined.

No critical points found

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Multivariable calculus

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?