#x,y,z# real positive numbers the smallest answer to ??

Question

#x^3+12((yz)/x)+16(1/(yz))^(3/2)= ???#
answers :
24 20 18 16 12

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Answer 1 · 2017-12-12T22:10:01

See below.

Making #yz = gamma^2# we have

#f(x,gamma) = x^3+12(gamma^2/x)+16 gamma^-3#

The stationary points for #f# are those such that

#f_x = 3 x^2 - (12 gamma^2)/x^2 = 0#
#f_gamma = 24 gamma/x-48 gamma^-4 = 0#

Solving for #x,gamma#

#x = 2^(2/3), gamma = 2^(1/3)#

At this point we have

#f(2^(2/3),2^(1/3)) = 24#