Extrema of multivariable function

[haroldholt]

Hi

I'm studying for a calculus exam and I'm a little stuck on finding the extrema for multivariable functions.

For the particular question I'm trying to do now I need to find and classify the extrema for the function f(x,y) = (4x^2)(e^y) - 2x^4 - e^4y.

I can find the first derivatives, being fx = 8xe^y - 8x and fy = (4x^2)(e^y) - 4e^4y and I know I have to let them be equal to 0 to find where the extrema are located but I'm not sure how to do that. I guess it's just the exponentials that are throwing me off.

Any help would be appreciated.

 

[D H]

You have done well so far. The next step is to simultaneously set those partial derivatives to zero. So try that and show us the results.

 

[haroldholt]

Thanks

Well it's 0 = 8xe^y - 8x and 0 = (4x^2)(e^y) - 4e^4y. But I'm not sure where to go from there. I'm not sure what to do with the exponentials, I know they can never equal zero, but I'm not sure what that means for my equations.

 

[D H]

Simplify the f_x=0 equation.

 

[haroldholt]

0 = 8x(e^y - 1). But I still don't know what to do with the exponential.

 

[D H]

Any equation of the form a*b=0 tells you that either a=0 or b=0 (or both). This is obviously of that form. x=0 doesn't lead to any extrema. (Why not?) What happens when you set e^y-1 to zero?

 

[haroldholt]

Thanks for that. I now realize that y can be set to zero in the equation 8x(e^y - 1). And then if you sub y = 0 into the other equation you get 1 and -1 which yields the points (-1,0) and (1,0) (which is the answer in the back of the book ). And if x = 0 then the other equation doesn't work. Thanks heaps. I now see how stupid I was initially lol.

 

[uiulic]

I wish when I learned calculus (or any other courses) I could have such instruction.