3. Consider this function:4.x + aif x < 3f (x) = { 3.x2 – 7.x – 2a if x > 323 – 46if x = 3.Suppose I tell you that f(x) is continuous at x = 3. Find a and b.

the function is continuous at x=3 therefore, limx→3-fx=limx→3+fx=f3 now…

Answered: 3. Consider this function: 4.x + a if x…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

3. Consider this function:
4.x + a
if x < 3
f (x) = { 3.x2 – 7.x – 2a if x > 3
23 – 46
if x = 3.
Suppose I tell you that f(x) is continuous at x = 3. Find a and b.

Expert Solution

Step 1

the function is continuous at $x = 3$

therefore, $l i m_{x \to 3^{-}} f (x) = l i m_{x \to 3^{+}} f (x) = f (3)$

now

$\begin{array}{rcl} l i m_{x \to 3^{-}} f (x) & = & l i m_{x \to 3^{-}} x^{2} - 4 x + a \\ = & 3^{2} - 4 (3) + a \\ = & 9 - 12 + a \\ = & - 3 + a \end{array}$

Step 2

$\begin{array}{rcl} l i m_{x \to 3^{+}} f (x) & = & l i m_{x \to 3^{+}} 3 x^{2} - 7 x - 2 a \\ = & 3 (3^{2}) - 7 (3) - 2 a \\ = & 27 - 21 - 2 a \\ = & 6 - 2 a \end{array}$

$\begin{array}{rcl} l i m_{x \to 3^{-}} f (x) & = & l i m_{x \to 3^{+}} f (x) \\ - 3 + a & = & 6 - 2 a \\ - 3 - 6 & = & - 2 a - a \\ - 9 & = & - 3 a \\ \frac{- 9}{- 3} & = & a \\ 3 & = & a \end{array}$

therefore value of a = 3

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