Question

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

Answer 1

`x = (n-1/(4))pi`
`n=0, +-1,+-2.."integer"`

Explanation:

`y = cosx - sinx`

Take the derivative wrt x and set it to zero.

`dy/dx = -(sinx + cosx) = 0`

`rArr sinx = - cosx`
`rArr sinx/cosx = - 1`

`rArr tan(x) = - 1`
`x = tan^(- 1)(-1) = npi - pi/4 =(n-1/(4))pi`
`x = (n-1/(4))pi`
where `n=0, +-1,+-2.."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]}