If the function \(f(x) = \begin{cases} ax + 2b, & x \lt 1\\ 7, & x = 1\\ 3ax - b, & x \gt 1\end{cases}\) is continuous at \(x = 1\), then the values of a and b are
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(a = 3,\ b = 2\)
For continuity at x = 1, both one-sided limits must equal f(1) = 7.
\( \displaystyle \lim_{x \to 1^-} f(x) = a + 2b = 7 \)
\( \displaystyle \lim_{x \to 1^+} f(x) = 3a - b = 7 \)
From the second equation, \(b = 3a - 7\). Putting this in the first:
\(a + 6a - 14 = 7\)
\(7a = 21\)
\(a = 3\)
\(b = 9 - 7 = 2\)
So a = 3 and b = 2.