What are the values and types of the critical points, if any, of #f(x,z) = x^4 + 15z^2 + 2xz^2 - 456z^2#?

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Answer 1 · 2017-08-18T15:07:28

There is only a relative minimum at #=(0,441/2)#

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We calculate the partial derivatives of the function

#f(x,z)=x^4+15z^2+2xz^2-456z^2#

#=x^4+2xz^2-441z^2#

#f_x=4x^3+2z^2#

#f_z=4xz-882z#

The critical points are when #f_x=4x^3+2z^2=0#, #=>#, #x=0# and #z=0#

#f_z=4xz-882z=0#

#=2z(2x-441)=0#

#=>#, #z=0# and #x=441/2#

#f_( x x)=12x^2#

#f_(z z)=4x-882#

#f_(x z)=4z#

#f_(zx)=4z#

#D(x,z)=f_( x x)f_(z z)-f_(x z)^2#

#D(0,441/2)=12x^2(4x-882)-16z^2#

#=12*(441/2)^2-0>0#

#f_x(0,441/2)=4*(441/2)^3>0#

#D(0,0)=12x^2(4x-882)-16z^2=0#

This test is inconclusive

There is only a relative minimum at #=(0,441/2)#