Let be a non-zero constant

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

Problem

Let

a

be a non-zero constant. A curve is given by

x = t^{3} - 3at

y = t^{4} - 4at^{2} + 2a^{2}

, where

t \in \mathbb{R}

. (i) Obtain expressions for

\dfrac{dy}{dx}

and

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

in terms of

t

and

a

. (ii) Show that the curve possesses points of inflection only when

t^{2} = a

, and determine the corresponding coordinates in terms of

a

. (iii) For

a>0

, find the equation of the tangent to the curve at each point of inflection and state whether the curve changes from concave upward to concave downward or vice-versa at those points.

Related theory

Curve Analysis
Advanced Differentiation
Implicit and Parametric Differentiation
Derivatives of parametric functions
Continuity and Differentiability

Browse the problem library