The curve given implicitly by

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

The curve given implicitly by \,

x^{\tfrac{2}{3}}+y^{\tfrac{2}{3}}=4

forms a closed, rounded diamond shape called an astroid. Let

y=f(x)

denote the upper half of the astroid (where

y\ge 0

). Answer the following for

-8\le x\le 8.

(a) Derive an explicit expression for

\dfrac{dy}{dx}

in terms of

x

and

y

. (b) Write

\dfrac{dy}{dx}

entirely in terms of

x

by substituting

y=f(x)

from the original relation. (c) Determine all values of

x

in the interval

[-8,8]

for which

\dfrac{dy}{dx}=0.

Give the corresponding points on the curve. (d) Determine all values of

x

in the interval

[-8,8]

for which

\dfrac{dy}{dx}

does not exist but

f

is still defined. Explain geometrically what happens to the curve at each of these

x

–values. (e) Compile your results from parts (c) and (d) to state every critical point of the function

y=f(x)

[-8,8]

. Justify that no other critical points exist.