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- Homework Statement
- Using the ODE, show that the solution has a local minimum at x = 0.

- Relevant Equations
- First derivative test

The ODE given to us is y' = xcosy. I am having a bit of trouble when it comes to solving this problem. We are supposed to show that the solution has a local minimum at x = 0 with the hint to think of the first derivative test. However, I am only really familiar with the first derivative test when it comes to a function like f(x) not y' with two variables.

In order to solve this would be have to first take f(x,y) = xcosy and then solve the partial derivatives f

Any help would be appreciated as I'm a bit confused about how to approach this, thank you.

In order to solve this would be have to first take f(x,y) = xcosy and then solve the partial derivatives f

_{x}(x,y) and f_{y}(x,y), then equate them both to zero, solve for x and y to obtain a critical point, and finally calculate the second partial derivatives to prove there is a local minimum at x = 0? Or is there a much more efficient way to do this?Any help would be appreciated as I'm a bit confused about how to approach this, thank you.