Problem Statement
Find the values of $a$ and $b$ such that the function $f(x)$ defined by:
$f(x) = \begin{cases} 5, & x \le 2 \\ ax + b, & 2 < x < 10 \\ 21, & x \ge 10 \end{cases}$
is a continuous function.
Verified Solution & Marking Scheme
Continuity Condition at x = 2
For $f(x)$ to be continuous at $x = 2$:
$\lim_{x \to 2^-} f(x) = \lim_{x \to 2^+} f(x) = f(2)$
- $\lim_{x \to 2^-} f(x) = 5$
- $\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (ax + b) = 2a + b$
Hence, $2a + b = 5$ ...(1)
Continuity Condition at x = 10
For $f(x)$ to be continuous at $x = 10$:
$\lim_{x \to 10^-} f(x) = \lim_{x \to 10^+} f(x) = f(10)$
- $\lim_{x \to 10^-} f(x) = \lim_{x \to 10^-} (ax + b) = 10a + b$
- $\lim_{x \to 10^+} f(x) = 21$
Hence, $10a + b = 21$ ...(2)
Solve Equations (1) and (2) Simultaneously
Subtracting (1) from (2):
$(10a + b) - (2a + b) = 21 - 5$
$8a = 16 \implies a = 2$
Substituting $a = 2$ into (1):
$2(2) + b = 5 \implies 4 + b = 5 \implies b = 1$.