Question

If \( f(x) = \begin{cases} b, & 0 \leq x < 1 \\ x + 3, & 1 < x \leq 2 \\ 4, & x = 1 \end{cases} \) , then the value of \( (a, b) \) for which \( f(x) \) cannot be continuous at \( x = 1 \) is

• A. (2, 2)
• B. (3, 1)
• C. (4, 0)
• D. (5, 2)

Hint:

If $f(x)=\{$, then the value of (a, b) for which $f(x)$ cannot be continuous at $x=1$ is

Answer 1

The Correct Option is D

Step 1: Concept
For $f(x)$ to be continuous at $x=1$, LHL = RHL = $f(1)$.
Step 2: Analysis
LHL: $lim_{x\rightarrow1^-} (ax^2 + b) = a + b$. RHL: $lim_{x\rightarrow1^+} (x+3) = 4$. $f(1) = 4$.
Step 3: Evaluation
Continuity requires $a + b = 4$. Check the options: (a) $2+2=4$, (b) $3+1=4$, (c) $4+0=4$.
Step 4: Conclusion
For option (d), $5+2=7 \ne 4$. Thus, $f(x)$ cannot be continuous for $(a, b) = (5, 2)$.
Final Answer: (d)