Is #f(x)=(x-2)^2/(x+1)# increasing or decreasing at #x=2#?
1 Answer
Mar 3, 2016
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]}
