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    Find critical points and test for relative extrema.

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    List the critical points for which the second partials test fails.

    f(x,y)=x^3+y^3-6x^2+9y^2+12x+27y+19

    © BrainMass Inc. brainmass.com February 4, 2021, 6:29 pm ad1c9bdddf
    https://brainmass.com/math/calculus-and-analysis/find-critical-points-test-relative-extrema-9020

    Solution Preview

    First we find the critical points of the function f(x,y):
    f1(x,y) is the first derivative with respect to x and f2(x,y) is the first derivative with respect to y. So:

    f1(x,y) := 3*x^2-12*x+12
    f2(x,y) := 3*y^2+18*y+27

    We have to solve the system of f1=0 and f2=0 and find all the common (x,y) roots of both. As you can see f1 has no y element and f2 has no x component, so you can solve them separately.

    f1=0 -> x=2, x=2 (repeated)
    f2=0 -> y=-3, y=-3 (repeated)
    ...

    Solution Summary

    Critical points are found and a function is tested for extrema. The second partial test fails are listed for a function.

    $2.19

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