Mathematics

Problem

The implicit relation

x^{2}y+y^{3}=6x

defines a curve in the

xy

-plane. a. Compute

\dfrac{dy}{dx}

in terms of

x

and

y

, clearly indicating each differentiation step. b. Obtain an expression for the second derivative

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

in terms of

x

and

y

. Simplify fully. c. Evaluate

\dfrac{dy}{dx}

and

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

at the point

(2,1)

. Based on these values, discuss the concavity of the curve at that point and describe the behavior of the curve near

(2,1)