Question

For what values of x is dy/dx zero and undefined?

`y = "x^2 - 3x + 1"/"x+2"`

Answer 1

`dy/dx` is zero for `x =-2 pm sqrt(11)`, and `dy/dx` is undefined for `x=-2`

Explanation:

Find the derivative:
`dy/dx = (d(x^2 - 3x + 1))/dx 1/(x+2) + (x^2 - 3x + 1)(d)/(dx)(1/(x + 2))`
`=(2x-3)/(x+2) - (x^2 - 3x + 1)1/(x + 2)^2`
`=((2x-3)(x+2) - (x^2 - 3x + 1))/(x + 2)^2`
`= (2x^2 - 3x + 4x -6 - x^2+3x-1)/(x + 2)^2`
`= (x^2 + 4x -7)/(x + 2)^2`
by the product rule and various simplifications.

Find zeros:
`dy/dx=0` if and only if `x^2 + 4x -7=0`.
The roots of this polynomial are
`x_{1,2} = (1/2)(-4 pm sqrt(4^2 - 4(-7))) = -2 pm sqrt(11)`,
so `dy/dx=0` for `x = -2 pm sqrt(11)`.

Find where `dy/dx` is undefined:
Since division by `0` is not allowed, `dy/dx` is undefined where `(x + 2)^2=0`, that is, where
`x=-2`.