Define the function F(x)= cases |mx+b|, & x 1

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Define the function

F(x)=\begin{cases} |mx+b|, & x\le 1,\\ k(x-1)^3+c, & x>1, \end{cases}

where

m,\,b,\,k,

and

c

are real constants. Assume

m>0

and

b<0

so that

|mx+b|=-(mx+b)

for

x=1

. (1) Part 1: (a) Determine all conditions that must hold among

m,\,b,\,k,

and

c

for

F

to be differentiable at

x=1

. Present each condition as an explicit algebraic equation. (2) Part 2: (b) Given in addition that

F(1)=4

and that the common derivative at

x=1

equals

6

, find the unique set of values

(m, b, k, c)

satisfying both the differentiability conditions from part (a) and these numerical requirements. (3) Part 3: (c) Confirm that your solution satisfies the original assumption on the sign of

|mx+b|

x=1

. Then discuss whether any other choice of signs for

m

and

b

could yield a differentiable function under the same numerical conditions. Provide a concise mathematical justification.

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