Implicit derivative - Graphic

Question

Find $\left(\Large\frac{dy}{dx}\right)_{x=1}$ and $\left(\Large\frac{d^2y}{dx^2}\right)_{x=1}$, if

$$x^2 -2xy +y^2 +x+y -2 = 0$$

Using the obtained results, show aproximately the proportions of the given curve in the neighbourhood of $x=1$

Obtained Results:

$\left(\Large\frac{dy}{dx}\right)_{x=1}= 3 $ (at $y=0$) or $\left(\Large\frac{dy}{dx}\right)_{x=1} = -1$ (at $y=1$)

$\left(\Large\frac{d^2y}{dx^2}\right)_{x=1} = 8 $ (at $y=0$) or $\left(\Large\frac{d^2y}{dx^2}\right)_{x=1} = -8$ (at $y=1$)

I'm not sure about the right procedure to plot this curve. Having the signals of the derivatives, we can check the monotonicity and its concavity. But, I'm kinda confused

Thanks!

Answer 1

Regarding the result you've achieved, we can plot that proportion as follows:

Answer 2

Some hints:

You have presumably already decided that the 2 points on the curve that we are interested in are $P(1,0)$ and $Q(1,1)$.

Plot these two points, and then use the fact that the gradient of the curve as it passes through $P$ is $1$, so that you can sketch the tangent to the curve at $P$. Do the same thing at point $Q$.

You then use the information from the second derivatives to decide which side of the tangent the curve lies on at the two points.