Answer
At $P(1,0)$, the graph has a vertical tangent $x=1$.
Work Step by Step
$$y^3+\cos xy=x^2$$
The graph of the curve is enclosed below.
a) As the point $(1,0)$ lies in the curve, $P$ satisfies the equation.
b) Find $dy/dx$ using implicit differentiation: $$3y^2+(-\sin xy)\frac{d}{dy}(xy)=2x$$ $$3y^2-\sin xy(y+x\frac{dy}{dx})=2x$$ $$3y^2-y\sin xy-x\sin xy\frac{dy}{dx}=2x$$ $$x\sin xy\frac{dy}{dx}=3y^2-y\sin xy-2x$$ $$\frac{dy}{dx}=\frac{3y^2-y\sin xy-2x}{x\sin xy}$$
- For $P(1,0)$: $$\frac{dy}{dx}=\frac{3\times0^2-0\times\sin(1\times0)-2\times1}{1\times\sin(1\times0)}=\frac{0-0-2}{1\times\sin0}=-\frac{2}{0}$$
$dy/dx$ is not defined for $P(1,0)$.
c) Since $dy/dx$ is not defined for $P(1,0)$ and from the graph of the curve, we see that at $(1,0)$, the curve has a vertical tangent $x=1$.
