Mathematics

Problem

A function

f

is defined by

f(x)=\begin{cases} ax^{2}+bx+3, & x\le 2,\\ \ln(x)+c, & x>2. \end{cases}

The real constants

a

b

, and

c

are unknown. a. Part 1: Determine the conditions on

a

b

, and

c

that guarantee

f

is continuous at

x=2

. Show all algebraic steps. b. Part 2: Determine the additional condition(s) required so that

f

is differentiable at

x=2

. Explicitly state any equations you obtain and solve for the constants. c. Part 3: Assuming the constants satisfy the differentiability condition, compute

f'(2)

. d. Part 4: Suppose the constants satisfy the continuity condition from part (a) but fail to satisfy the differentiability condition from part (b). Evaluate

\displaystyle \lim_{x\to 2} \frac{f(x)-f(2)}{x-2}

and interpret its value with respect to the existence of

f'(2)

. Provide justification based on one–sided limits.

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