If the function \(f(x) =\begin{cases}ax+2&,x\le 1\\x^2+3x+b&,x\gt 1\end{cases}\) is differentiable at \(x = 1\), then the value of \((2a + b)\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 13
A function differentiable at x = 1 is also continuous there, and its left and right derivatives at x = 1 are equal.
Continuity at x = 1 (left value = right limit):
\(a(1) + 2 = 1 + 3 + b\)
\(a - b = 2\)
Equal derivatives at x = 1:
\(\text{Left derivative} = a\)
\(\text{Right derivative} = 2(1) + 3 = 5\)
\(a = 5\)
Then
\(b = a - 2 = 3\)
\(2a + b = 10 + 3 = 13\)