There are 30 students in the gym. If there are at least 16 girls, write an inequality to represent the number of boys in the gym.

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jsjsjsskakwkowwj

05.01.2021

Mathematics

$z=x^2+2y^2+6x-4y+22\\
z_x'=2x+6\\
z_y'=4y-4\\\\
2x+6=0\\
4y-4=0\\\\
2x=-6\\
4y=4\\\\
x=-3\\
y=1\\\\

$

We have a stationary point (-3,1)

$z_{xx}''=2\\
z_{xy}''=z_{yx}=0\\
z_{yy}''=4$

$2\cdot4-0^2=8 \ \textgreater \ 0$ therefore there exist an extremum at this point.

$z_{xx}''\ \textgreater \ 0$ therefore the extremum indeed is the minimum.

$z(-3,1)=(-3)^2+2\cdot1^2+6\cdot(-3)-4\cdot1+22=9+2-18-4+22\\
z(-3,1)=11$

So, the minimum is equal to 11.

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