Question

Is `f(x)=(x-2)^2/(x+1)` increasing or decreasing at `x=2`?

Answer 1

Neither.

Explanation:

To test if the function is increasing / decreasing at x = 2 , require to evaluate f'(2).

• If f'(2) > 0 then f(x) is increasing at x = 2

• If f'(2) < 0 then f(x) is decreasing at x = 2

differentiate using`color(blue)" Quotient rule and chain rule "`

`f'(x) = ((x+1) d/dx(x-2)^2 - (x-2)^2 d/dx(x+1))/(x+1)^2`

`=((x+1).2(x-2) d/dx(x-2) -(x-2)^2 .1)/(x+1)^2`

`=( 2(x+1)(x-2) - (x-2)^2)/(x+1)^2`

f'(2) = 0/9 = 0

`rArr "neither increasing nor decreasing "`

f'(2) = 0 indicates a stationary point at x = 2
Here is the graph of the function
graph{(x-2)^2/(x+1) [-10, 10, -5, 5]}