We have the function `y = x^3/(x+1)^2`
a) `x` intercepts are the values of `x` when `y =0`.
The only value of `x` for which this is true is `x = 0`
b) `y` intercepts are the values of `y` when `x =0`. When `x=0`, `y=0`.
Since this is a function (one-one), that is the only `y` intercept
c) To find the extrema, differentiate the function
`(dy)/(dx) = (3x^2)/(x+1)^2 + x^3((-2))/(x+1)^3 = (3x^2(x+1) - 2x^3)/(x+1)^3`
`= (3x^3 + 3x^2 - 2x^3)/(x+1)^3 = (x^2(x + 3))/(x+1)^3`
The extrema are at `x` such that `(dy)/(dx) = 0`
So the extrema are at `x=0` and `x = -3`
At `x=-3`, `y = (-3)^3/(-3 +1)^2 = -27/4`
The coordinates of the extrema are `(0,0)` and `(-3,-27/4)`
To find whether they are maxima, minima or points of inflection check the sign of `(d^2y)/(dx^2)`
Now, `(d^2y)/(dx^2) = (d/(dx))(dy)/(dx) = (3x^2+6x)/(x+1)^3 +x^2(x+3)((-3))/(x+1)^4`
When `x=0` this equals 0 `implies` point of inflection at `(0,0)`
When `x=-3` this equals `(27-18)/(-8) = -9/8` `implies` maximum at `(-3,-27/4)`
d) A vertical asymptote is a value of `x` such that `y -> +-oo`.
As `x -> -1`, `y -> -1/0` so `y -> -oo`
So there is a vertical asymptote at `x =-1`
e) A horizontal asymptote is a value of `y` such that `x -> oo`
` `` ` ` `` ` ` ` ` ` ` `` `` `` ` There are no horizontal asymptotes
f) A slant asymptote is when `(x,y)` tends to a line other than` ` a horizontal or vertical one.` `
Rewrite `y =x^3/(x+1)^2` as `y =x/(1/x^2(x+1)^2) = x/(1+1/x)`
Now, as `x -> oo`, `1/x -> 0` and `y -> x/1 = x` (similarly for `x -> -oo`)
Therefore the line `y=x` is a slant asymptote
g) Points of inflection are when `(d^2y)/(dx^2) = 0`
Here this is when `(3x^2 + 6x)/(x+1)^3 +x^2(x+3)(-3)/(x+1)^4 = 0`
ie when `((3x^2 + 6x)(x+1) - 3x^2(x+3))/(x+1)^4 = 0`
`implies 3x^3 + 9x^2 + 6x - 3x^3 - 9x^2 = 0` `implies 6x =0`
There is only one point of inflection, at `(0,0)`
h) The graph is continuous to the left from `x = -1` and continuous to the right from `x = -1`. There is a singularity at `x = -1`.
The graph is very similar to `y = x` except near the origin where there is a discontinuity, a maxmimum and a point of inflection.
sketch of graph y = x^3/(x+1)^2 looks like this