Need help with Lagrange multiplier question?

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Answer 1 · 2018-04-15T22:46:48

See below.

#L(x,y,lambda) = 5x^2+6y^2-xy + lambda(x+2y-24)#

#grad L = ((10x-y+lambda),(-x+12y+2lambda),(x+2y-24))=((0),(0),(0))#

Now solving for #x,y,lambda#

#x=6, y = 9, lambda =-51# ----(*)

The qualification is done over #f @ g# or

#f@g = 7x^2-84x+864#

hence #x = 6, y = 9# as determined in (*) is a local minimum.