Find the D value of critical point and the type of critical point

1. The problem statement, all variables and given/known data



The function f(x,y) = [e^(-y^2)]cos(4x) has a critical point (0,0)



2. Relevant equations



Find the D value at the critical point. What type of critical point is it? (max, min, saddle or none)



3. The attempt at a solution



I know that to find the D value I must compute the partial derivatives (fox, fey, fxy and fyx)



fx = [-4e^(-y^2)]sin(4x)



fy = [-2ye^(-y^2)]cos(4x)



fxx = [-16e^(-y^2)](cos(4x)



fyy = [4(y^2)e^(-y^2)]cos(4x)



fxy = [8ye^(-y^2)]sin(4x)



fxy = [8ye^(-y^2)]sin(4x)



D = (fxx)(fyy) - (fxy)^2

= [-16e^(-y^2)](cos(4x)[4(y^2)e^(-y^2)]cos(4x) - {[8ye^(-y^2)]sin(4x)}^2



evaluated at the critical point (0,0)



= (-16)(0) - (0)(0)

= 0



This means that the 2nd derivative test gave no info. But somehow it's not the right answer. Did I find the correct value of D? Is my approach correct?



Thanks

1. The problem statement, all variables and given/known data







2. Relevant equations







3. The attempt at a solution

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