I am asked to find the equations of the tangent lines at three different points for the following function:
$$y^{5}-y-x^{2}=-1$$
I am provided with a graph/sketch of the function and asked to find the tangent lines to the three different points at $x=1$. I found the following with implicit differentiation:
$$\frac{dy}{dx}=\frac{2x}{5y^{4}-1}$$
My problem is finding the points where the curve intersects $x=1$. It is pretty clear from the graph (though not explicitly marked) that the points will be $y=-1$,$y=0$, and $y=1$. Wolfram Alpha confirms this http://www.wolframalpha.com/input/?i=y%5E5-y-x%5E2%3D-1+and+x%3D1. I just don't know if seeing them visually is enough or if I have to derive the intersection points mathematically.
Thank you very much.
Best Answer
Well, have you tried solving? When $x=1$, you have $y^5 - y -(1)^2 = -1$, which means $y^5 = y$, hence either $y=0$ or $y^4 = 1$, which means $y=\pm 1$. Satisfied?
Hope that helps =)