Let `f(x)={a x^2+1,x >1`;
`x+1/2,x<=1.t h e n ,f(x)` is derivable at `x=1,` if
`a=2`
(b) `a=1`
(c) `a=0`
(d) `a=1/2`

Explanation

To determine the value of \( a \) for which the function \( f(x) \) is differentiable at \( x = 1 \), we need to ensure that the function is both continuous and has equal left-hand and right-hand derivatives at that point.

### Step 1: Define the function
The function is given as:
\[
f(x) =
\begin{cases}
a x^2 + 1 & \text{if } x > 1 \\
...