If the function $f(x) =\left\{\begin{matrix}ax +2&,x≤1\\x2+3x+b&,x>1\end{matrix}\right.$ is differentiable at $x = 1$, then the value of (2a + b) is

13

The correct answer is Option (1) → 13

$f(x)=\begin{cases} ax+2, & x\le1\\ x^{2}+3x+b, & x>1 \end{cases}$

For differentiability at $x=1$, function must be continuous and derivatives equal.

Continuity at $x=1$:

$f(1^-)=a(1)+2=a+2,\quad f(1^+)=1^{2}+3(1)+b=4+b$

So, $a+2=4+b\Rightarrow a-b=2$ … (1)

Equal derivatives at $x=1$:

$f'(x)=\begin{cases} a, & x\le1\\ 2x+3, & x>1 \end{cases}$

At $x=1$, $a=2(1)+3=5$

Substitute in (1): $5-b=2\Rightarrow b=3$

Required $(2a+b)=2(5)+3=13$