The function is #y=xsqrt(16-x^2)#
We need the domain of #y#
#16-x^2>=0#, #=>#, #x^2>=16#
Therefore the domain is #x in (-4,4)#
We need the derivative is
#(uv)'=u'v+uv'#
#u(x)=x#, #=>#, #u'(x)=1#
#v(x)=sqrt(16-x^2)#, #=>#, #v'(x)=(-2x)/(2sqrt(16-x^2))=-x/(sqrt(16-x^2))#
So,
#dy/dx=sqrt(16-x^2)-x^2/(sqrt(16-x^2))=(16-x^2-x^2)/(sqrt(16-x^2))#
#=(16-2x^2)/(sqrt(16-x^2))#
The critical points are when #dy/dx=0#
#(16-2x^2)/(sqrt(16-x^2))=0#
That is
#16-x^2=0#, #=>#, #x^2=8#, #=>#, #x=+-sqrt8#
We can build the variation chart
#color(white)(aaaa)##Interval##color(white)(aaaa)##(-4,-sqrt8)##color(white)(aaaa)##(-sqrt8, sqrt8)##color(white)(aaaa)##(sqrt8,4)#
#color(white)(aaaa)##dy/dx##color(white)(aaaaaaaaaaaaa)##-##color(white)(aaaaaaaaaaaa)##+##color(white)(aaaaaaaaa)##-#
#color(white)(aaaa)##y##color(white)(aaaaaaaaaaaaaaa)##↘##color(white)(aaaaaaaaaaaa)##↗##color(white)(aaaaaaaaa)##↘#
graph{xsqrt(16-x^2) [-16.02, 16.01, -8.01, 8.01]}