For the one-parameter family of functions defined for all real

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

For the one-parameter family of functions

f_{k}(x)=x^{3}+kx^{2}+4x

defined for all real

x

, let

k

be a real constant. (a) Compute

f_{k}''(x)

. (b) Determine all real values of

k

for which the graph of

f_{k}

is concave up for every

x\ge0

. Show complete reasoning. (c) For

k=-3

, find the intervals of concavity of

f_{k}

on the entire real line and identify any points of inflection.