For , consider the function…

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

For

k \in \mathbb{R}

, consider the function

y = \bigl(x^2 + kx + 1\bigr)^3

. (i) Find

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

in terms of

x

and

k

. (ii) Determine the value of

k

for which

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

equals zero at

x = 1

. (iii) For the value of

k

obtained in part (ii), decide whether the curve has an inflection point at

x=1

, giving reasons for your answer.

Related theory

Curve Analysis
Higher-Order Derivatives
Advanced Differentiation
Differentiation Rules
Second order derivatives
Continuity and Differentiability

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