The curve defined implicitly by has a critical point at

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

The curve defined implicitly by

x^{4}-y=0

has a critical point at

(0,0)

. At that point

\dfrac{dy}{dx}=0

and

\dfrac{d^{2}y}{dx^{2}}=0

. Which statement best describes

(0,0)

? (a)

(0,0)

is a saddle point because the second derivative is zero. (b)

(0,0)

is a local minimum even though the second derivative test is inconclusive. (c) No conclusion can be drawn about

(0,0)

; additional analysis cannot classify it. (d)

(0,0)

is a local maximum because the second derivative is zero.