Consider the family of twice-differentiable functions defined by f_k(x)=x^ 4 -2kx^ 2…

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

Problem

Consider the family of twice-differentiable functions defined by

f_k(x)=x^{4}-2kx^{2}+k^{2}x,

where

k

is a real constant. (a) Derive expressions for

f_k'(x)

and

f_k''(x)

in terms of

k

. (b) For a fixed

k

, determine all critical points of

f_k'

. Express each

x

-coordinate in simplest form in terms of

k

. (c) For what real values of

k

does the graph of

f_k

possess exactly two distinct points of inflection? Show all reasoning, including an analysis of how the number of sign changes of

f_k''

depends on

k

. (d) Select one value of

k

that satisfies the condition obtained in part (c). For this specific value, compute the coordinates of the two inflection points and discuss the intervals on which

f_k

is concave upward and concave downward.

Related theory

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

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