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Find critical points and classify them as local maxima, local minima, saddle points, or none of these?
Let f(x,y)=e^(−x^(2))+6y^(3). Find critical points and classify them as local maxima, local minima, saddle points, or none of these.
1 Answer
- kbLv 75 years agoFavorite Answer
Critical points:
f_x = -2xe^(-x^2) and f_y = 18y^2.
Setting these equal to 0 yields the critical point (0, 0).
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Next, we classify this point with the Second Derivative Test.
f_xx = -2e^(-x^2) + -2x * -2xe^(-x^2) = (4x^2 - 2) e^(-x^2)
f_xy = 0
f_yy = 36y.
So, D = (f_xx)(f_yy) - (f_xy)^2 = 36y(4x^2 - 2) e^(-x^2).
Since D(0, 0) = 0, the Second Derivative Test is inconclusive.
However, we can check that (0, 0) is a saddle point by examining neighboring points to (0,0). (Note that f(0,0) = 1).
For instance for any small d>0, we have f(0,d) = 1 + 6d^3 > f(0,0), but
f(0,-d) = 1 - 6d^3 < f(0,0).
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I hope this helps!
