Find the relationship between a and b so that the function f defined by fx- a x-1- if x - 3- b x-3- if x-3 -is continuous at x-3-

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Relation between Continuity and Differentiability

Find the rela...

Question

Find the relationship between a and b so that the function f defined by

is continuous at x = 3.

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Solution

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a

b

f

f (x) = {\begin{matrix} a x + 1, & i f x \leq 3 b x + 3, & i f x > 3 \end{matrix}

x = 3

a

b

f

f (x) = {\begin{matrix} a x + 1, b x + 3, \end{matrix} \begin{matrix} i f x \leq 3 i f x > 3 \end{matrix}

x = 3

a

b

f (x)

f (x) =

{\begin{matrix} a x + 2, i f x \leq 4 b x + 4, i f x > 4 \end{matrix}

x = 3

⎧ ⎨ ⎩ \begin{matrix} 2 x, i f x < 0 0, i f 0 \leq x \leq 1 4 x, i f x > 1 \end{matrix}

f (x) = \{\begin{cases} a x + 1, & if x \leq 3 \\ b x + 3, & if x > 3 \end{cases}

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