Question

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

Answer 1

So the critical point is `x=0`

Explanation:

`y= cos(x/(x+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))`( use chain rule of differentiation)

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

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

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

So the critical point is `x=0`