How do you use the first and second derivatives to sketch # f(x)=(x+2)/(x-3)#?

1 Answer
Jan 8, 2017

graph{(x+2)/(x-3) [-15.58, 24.42, -8.16, 11.84]}

Explanation:

#f(x) = (x+2)/(x-3)#

#f'(x) = frac ( x-3 -x -2) ((x-3)^2) = -5/((x-3)^2)#

#f''(x) = 10/((x-3)^3)#

We can now analyse the behaviour of the derivatives to sketch the function:

(1) #f'(x) < 0# everywhere in its domain #RR -{3}#, so #f(x)# is strictly decreasing and has no local extrema.

(2) for #x < 3#, #f''(x) < 0# so #f(x)# is concave down in #(-oo,3)#

(3) for #x > 3#, #f''(x) > 0# so #f(x)# is concave up in #(3,+oo)#