Question

y= 2x/ (1-x)
y'' = 4 / (1-x)^3

I need to find the concavity and the point of inflection
so I need to use the second derivative
but x=1 is undefined, so does it mean there is no concavity nor point of inflection?

Answer 1

no. depending on the sign of f'' on the left/right of x=1, will mean concave up/down on the left/right of x=1

Answer 2

and you're right, there is no inflection point. but there might be a change in concavity as you go from left of x=1 to the right of x=1.

Answer 3

it is + in the intervals of ( - infinity , 1) and -ve in ( 1, + infinity)
so there is point of inflection?

Answer 4

but then when i plug in x=1, it is undefined

Answer 5

so there is still concavity even there is no point of inflection?

Answer 6

take this for example. the function is undefined at x=c but on the left of x=c, the function is concave down. but to the right of x=c, the function is concave up.