Mathematics

Problem

The relation

x^{2}+xy+y^{2}=7

defines

y

implicitly as a differentiable function of

x

near the point

(1,2)

. a. Obtain an expression for

\dfrac{dy}{dx}

in terms of

x

and

y

. Simplify as much as possible. b. Differentiate your result from part (a) with respect to

x

to obtain an explicit formula for

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

in terms of

x

and

y

. Show all algebraic steps. c. Evaluate

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

at the point

(1,2)

and state whether the curve is concave upward or concave downward there. Justify your conclusion.