The set of points satisfying with forms a closed curve in the plane.

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

The set of points

(x,y)

satisfying

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

with

y>0

forms a closed curve in the plane. (a) Obtain an explicit expression for

\dfrac{dy}{dx}

in terms of

x

and

y

. (b) Confirm that

(0,1)

lies on the curve and compute

\left.\dfrac{dy}{dx}\right|_{(0,1)}

. (c) Find

\dfrac{d^{2}y}{dx^{2}}

in terms of

x

and

y

, and then evaluate

\dfrac{d^{2}y}{dx^{2}}

(0,1)

. (d) Using your result from part (c), determine whether the curve is concave upward or concave downward at

(0,1)

. Provide justification based on the sign of the second derivative.