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
Answer & explanation
Correct answer: option 1
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 $(2a+b)=2(5)+3=13$