Mathematics

Problem

For a fixed real number

k>0

, consider the family of functions

f_k(x)=|x|^{k}.

(a) For

x\ne0

, establish a simplified formula for the difference quotient

\dfrac{f_k(x+h)-f_k(x)}{h}

and describe how its behavior depends on

k

. (b) Evaluate

\displaystyle\lim_{h\to0} \dfrac{f_k(h)-f_k(0)}{h}

explicitly in terms of

k

. (c) Determine all values of

k

for which

f_k

is differentiable at

x=0

. Provide a rigorous justification rooted in the limit obtained in part (b). (d) When the derivative at

0

exists, state its exact value and discuss how the smoothness of

f_k

near the origin changes qualitatively as

k

passes through the threshold found in part (c). a. Part (a) — General form of the difference quotient for

x\ne0

b. Part (b) — Limit evaluation at the origin c. Part (c) — Parameter values ensuring differentiability at

0

d. Part (d) — Value of the derivative and interpretation of smoothness