Question

Let \(f : R \to R\) be a function defined as
\(f(x) = \begin{cases}5 & ; x \le 1\\ a + bx & ; 1 \lt x \lt 3\\ b + 5x & ; 3 \le x \lt 5\\ 30 & ; x \ge 5\end{cases}\)
Then, f is

• A. continuous if \(a = 0\) and \(b = 5\)
• B. continuous if \(a = -5\) and \(b = 10\)
• C. continuous if \(a = 5\) and \(b = 5\)
• D. not continuous for any values of \(a\) and \(b\)

Hint:

Impose continuity at the break points \(x = 1, 3, 5\) and see whether the three conditions can hold together.

Answer 1

The Correct Option is D

Continuity at \(x = 1\): \(a + b = 5\). (i)
Continuity at \(x = 3\): \(a + 3b = b + 15 \Rightarrow a + 2b = 15\). (ii)
Continuity at \(x = 5\): \(b + 25 = 30 \Rightarrow b = 5\). (iii)
From (iii) and (i): \(a = 0\). Then (ii) gives \(0 + 10 = 10 \ne 15\), a contradiction.
So no values of \(a, b\) make \(f\) continuous.
Answer: (4) not continuous for any values of \(a\) and \(b\)