# Using an ODE to show a local minimum

ver_mathstats
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 fx(x,y) and fy(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.

Delta2

Homework Helper
Gold Member
Let me rewrite the ode to make some things more clear to you:
The ODE is $$y'(x)=x\cos(y(x))$$. So what can you say about ##y'(0)##(Hint: Apply the ODE for x=0).

ver_mathstats
Homework Helper
2022 Award
The sign of the second derivative depends on the value of $y(0)$, so I don't think you can conclude that it's a minimum if you aren''t given that information.

ver_mathstats and Delta2
ver_mathstats
The sign of the second derivative depends on the value of $y(0)$, so I don't think you can conclude that it's a minimum if you aren''t given that information.
We were given y(0) = 0, sorry told not to use initially but now apparently we can use it.

Last edited:
Homework Helper
Gold Member
We were given y(0) = 0, sorry told not to use initially but now apparently we can use it.
You need it to prove that ##y''(0)>0## hence it is a minimum. By differentiating the given ODE one time you end up with ##y''(x)=\cos(y(x))-xy'(x)\sin(y(x))##

ver_mathstats
ver_mathstats
You need it to prove that ##y''(0)>0## hence it is a minimum. By differentiating the given ODE one time you end up with ##y''(x)=\cos(y(x))-xy'(x)\sin(y(x))##
Thank you, this makes so much sense now! I appreciate it.