Maximizer/minimizer of multivariable function

[calc2302]

Question:

Does f[x, y] = [Sin(2x) + Sin(3y)] / [0.5 + x^2 + y^2] have a maximizer and/or minimizer?

Explain.

I don't believe it has a maximizer nor a minimizer but not sure how to explain this.

 

[Deleted member 4993]

Question:

Does f[x, y] = [Sin(2x) + Sin(3y)] / [0.5 + x^2 + y^2] have a maximizer and/or minimizer?

Explain.

I don't believe it has a maximizer nor a minimizer but not sure how to explain this.

I believe I have never come across the term. What does "maximizer and/or minimizer" of a function mean mathematically?

 

[calc2302]

like global maximum/minimum of a function

 

[Deleted member 4993]

Question:

Does f[x, y] = [Sin(2x) + Sin(3y)] / [0.5 + x^2 + y^2] have a maximizer and/or minimizer?

Explain.

I don't believe it has a maximizer nor a minimizer but not sure how to explain this.

What are the conditions for maxima/minima for a function with two variables?

 

[calc2302]

fx = 0 & fy = 0 would give stationary point (x, y)

Substitute into below:
If fxx * fyy − f^2 xy > 0 it is either a max or min.

If fxx <0 and fyy <0 it is a maximum
If fxx >0 and fyy >0 it is a minimum

 

[Deleted member 4993]

fx = 0 & fy = 0 would give stationary point (x, y)

Substitute into below:
If fxx * fyy − f^2 xy > 0 it is either a max or min.

If fxx <0 and fyy <0 it is a maximum
If fxx >0 and fyy >0 it is a minimum

Correct. So did you calculate at which points f_x = 0 and f_y=0?

 

[calc2302]

No that's where I am stuck

 

[Steven G]

Can you show us your work for finding f_x and f_y? I would try using the quotient rule for each.