Question

How do I find the critical points of this function?
sqrt(x)/x+4

Answer 1

\[Let y=\frac{ \sqrt{x} }{ x+4}\]
differentiate w.r.t.x
\[\frac{ dy }{ dx }=\frac{ \left( x+4 \right)\frac{ 1 }{ 2 }x ^{\frac{ -1 }{ 2 }}-x ^{\frac{ 1 }{ 2 }}*1 }{ \left( x+4 \right)^{2} }\]
\[\frac{ dy }{ dx }=\frac{ \frac{ x+4 }{ \ 2sqrt{x} }-\sqrt{x} }{ \left( x+4 \right)^{2} }\]
\[\frac{ dy }{ dx }=\frac{ x+4-2x }{ 2\sqrt{x} \left( x+4 \right)^{2}}\]
\[\frac{ dy }{ dx }=\frac{ -x+4 }{ 2\sqrt{x}\left( x+4 \right)^{2} }\]
critical points are given by dy/dx =0
-x+4=0
x=4