Question

Find extreme values f(x,y)= 3xy on 2x+y=8

Answer 1

Observe that $x=2$ is the only critical value

Find the second derivative of the function

$g'(x)=24-12x\Rightarrow g''(x)=0-12(1)=-12$

Find the sign of the second derivative at the critical value $x=2$

$x=2\Rightarrow g''(2)=-12<0$

Since the sign of the second derivative at the critical value is negative, so the given function has maximum value at $x=2$

To find the maximum value, find the value of $y$ by substituting $x=2$ in $y=8-2x$

$x=2\Rightarrow y=8-2(2)=8-4=4$

Find the maximum value of the given function by substituting $x=2,y=4$ in the given function

$f(x,y)=3xy$

Now $x=2,y=4\Rightarrow f(2,4)=3(2)(4)=24$

Therefore, Maximum value of the given function $=24$

Since $x=2$ is the only critical value and the function has maximum value or minimum value

at the critical values only, so the given function has no minimum value.

Hence, the extreme value (maximum value) of the given function = 24