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

Jul 30, 2016

So the critical point is $x = 0$

#### Explanation:

$y = \cos \left(\frac{x}{x + 1}\right)$
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.

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \sin \left(\frac{x}{x + 1}\right) . \frac{d}{\mathrm{dx}} \left(\frac{x}{x + 1}\right)$( use chain rule of differentiation)

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \sin \left(\frac{x}{x + 1}\right) \left(\frac{1 \left(x + 1\right) - x .1}{x + 1} ^ 2\right)$Use product rule of differentiation.

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \sin \left(\frac{x}{x + 1}\right) \left(\frac{1}{x + 1} ^ 2\right)$

Set dy/dx=0
$- \sin \frac{\frac{x}{x + 1}}{x + 1} ^ 2 = 0$
$\Rightarrow \sin \frac{\frac{x}{x + 1}}{{\left(x + 1\right)}^{2}} = 0$
$\sin \left(\frac{x}{x + 1}\right) = 0 \Rightarrow \frac{x}{x + 1} = 0 \Rightarrow , x = 0$

So the critical point is $x = 0$