Question

Determine \(a\) and \(b\) if \(f(x)\) is a continuous function, where

\[ f(x) = \begin{cases} 2, & x \le 3 \\ ax + b, & 3 < x < 5 \\ 5, & x \ge 5 \end{cases} \]

Hint:

Equate the left and right limits at x = 3 and x = 5 to get two equations in a and b.

Answer 1

Step 1: State the condition:
Each piece is continuous on its own interval. So only the points \( x = 3 \) and \( x = 5 \) need checking.
At each point, the left limit, right limit and value must agree.

Step 2: Continuity at x = 3:
Left limit is \( 2 \) and \( f(3) = 2 \). The right limit is \( 3a + b \).
\[ 3a + b = 2 \quad \cdots (1) \]

Step 3: Continuity at x = 5:
The left limit is \( 5a + b \). The right limit is \( 5 \) and \( f(5) = 5 \).
\[ 5a + b = 5 \quad \cdots (2) \]

Step 4: Solve the two equations:
Subtract (1) from (2). This gives \( 2a = 3 \), so \( a = \dfrac{3}{2} \).
Put this in (1): \( b = 2 - \dfrac{9}{2} = -\dfrac{5}{2} \).

Final Answer:
\[ \boxed{a = \dfrac{3}{2}, \quad b = -\dfrac{5}{2}} \]