Mathematics

Problem

The curve

C

is defined implicitly by the equation

e^{xy}+y=x^{2}

. a. Derive an explicit formula for

\dfrac{dy}{dx}

in terms of

x

and

y

. Present your work in a logically ordered sequence of steps. b. Determine all points on

C

where the tangent line is horizontal. Express each point’s coordinates as ordered pairs and justify that your list is complete. c. For

x>0

, decide whether the curve can possess a vertical tangent. Provide a mathematical argument that supports your conclusion.