Question

What are the values and types of the critical points, if any, of `f(x)=(2x^2+5x+5)/(x+1)`?

Answer 1

critical points are 0, -2, -1
Minima at (0,5), Maxima at (-2,-3), Discontinuous at x=-1

Explanation:

critical points are those at which f'(x) is =0, or for which f'(x) is not defined.
In this case f'(x)= `((4x+5)(x+1)-(2x^2 +5x+5))/ (x+1)^2`

= `( 2x^2 +4x)/(x+1)^2`

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

Critical points are therefore x=0, x=-2 and x= -1

For determining the types find f"(x)= `((4x+4)(x+1)^2 - 2(x+1)(2x^2 +4x))/(x+1)^4`

For x=0, f"(x) would be +ive, hence it is a minima at the point (0, 5)

For x= -2, f"(x) would be -ive, hence it is a maxima at the point (-2,-3)

At x=-1, f(x) does not exist. x=-1 is a vertical asymptote