Mathematics

Problem

For

k \in \mathbb{R}

, consider the function

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

. a. (i) Find

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

in terms of

x

and

k

. b. (ii) Determine the value of

k

for which

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

equals zero at

x = 1

. c. (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.