For a real parameter

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

Problem

p

, define

h_p:[0,\infty)\to\mathbb{R}

h_p(x)=x^{3}-3px

. (a) For a fixed value of

p

, determine all

x>0

for which

h_p

is increasing. Express your answer in terms of

p

and justify your reasoning. (b) Find all real values of

p

such that

h_p

is one-to-one on

[0,\infty)

. Provide a complete justification. (c) For every

p

satisfying the condition in part (b), prove that the inverse function

h_p^{-1}

is continuous on its domain. (d) Assume

p

satisfies part (b). Evaluate

\displaystyle \lim_{x\to h_p(0)^{+}} h_p^{-1}(x)

and interpret the meaning of this limit in terms of continuity of the inverse at the endpoint.