A family of quartic functions is given by

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

F_{p}(x)=x^{4}+p x^{3}+3x^{2}

, where

p

is a real parameter. (1) In terms of

p

, compute

F_{p}'(x)

and determine all critical points of

F_{p}

as functions of

p

. (2) Focus on the critical point at

x=0

. For which real values of

p

does

F_{p}

have a local minimum at

x=0

? Justify your answer thoroughly. (3) Determine all real values of

p

for which

F_{p}

has any local maximum on the real line. Identify the

x

-coordinate(s) of such maxima and provide justification. (4) For a value of

p

at which

F_{p}

possesses exactly one local extremum, classify that extremum and explain why no other local extrema exist for that parameter choice.