# How do you find the critical numbers of y = cos x - sin x?

Jan 5, 2018

$x = \left(n - \frac{1}{4}\right) \pi$
$n = 0 , \pm 1 , \pm 2. . \text{integer}$

#### Explanation:

$y = \cos x - \sin x$

Take the derivative wrt x and set it to zero.

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \left(\sin x + \cos x\right) = 0$

$\Rightarrow \sin x = - \cos x$
$\Rightarrow \sin \frac{x}{\cos} x = - 1$

$\Rightarrow \tan \left(x\right) = - 1$
$x = {\tan}^{- 1} \left(- 1\right) = n \pi - \frac{\pi}{4} = \left(n - \frac{1}{4}\right) \pi$
$x = \left(n - \frac{1}{4}\right) \pi$
where $n = 0 , \pm 1 , \pm 2. . \text{integer}$

You can also inspect the plot of the function to verify the critical values. Click on maxima and minima of the graph to find their x values and compare to the solution.

y = cosx-sinx plot:
graph{(cosx- sinx) [-10, 10, -5, 5]}