How do you find the critical numbers for cos (x/(x^2+1))cos(xx2+1) to determine the maximum and minimum?

1 Answer
Jul 30, 2016

So the critical point is x=0x=0

Explanation:

y= cos(x/(x+1))y=cos(xx+1)
Critical point : It is the point where the first derivative zero or it does not exist.
First find the derivative , set it to 0 solve for x.
And we need to check is there a value of x which makes the first derivative undefined.

dy/dx=-sin(x/(x+1)). d/dx(x/(x+1))dydx=sin(xx+1).ddx(xx+1)( use chain rule of differentiation)

dy/dx=-sin(x/(x+1))((1(x+1)-x.1)/(x+1)^2)dydx=sin(xx+1)(1(x+1)x.1(x+1)2)Use product rule of differentiation.

dy/dx=-sin(x/(x+1))((1)/(x+1)^2)dydx=sin(xx+1)(1(x+1)2)

Set dy/dx=0
-sin(x/(x+1))/(x+1)^2=0sin(xx+1)(x+1)2=0
rArrsin(x/(x+1))/((x+1)^2)=0sin(xx+1)(x+1)2=0
sin(x/(x+1))=0 rArr x/(x+1)=0 rArr ,x=0sin(xx+1)=0xx+1=0,x=0

So the critical point is x=0x=0