Consider the following for the next items that follow:

A function is defined by$f(x)=\left|\begin{array}{ccc} x+1 & 2 & 3 \\ 2 & x+4 & 6 \\ 3 & 6 & x+9 \end{array}\right|$ .

The function attains local minimum value at :

Let A be a matrix of order 3 × 3 and |A| = 4. If |2adj(3A)| = 2^α 3^β, then what is the value of (α + β)? 

What is the sum of the roots of the equation $\left|\begin{array}{ccc} 0 & x-a & x-b \\ 0 & 0 & x-c \\ x+b & x+c & 1 \end{array}\right|=0 $ ?

If a, b, c are in AP, then what is $\left|\begin{array}{lll} x+1 & x+2 & x+3 \\ x+2 & x+3 & x+4 \\ x+a & x+b & x+c \end{array}\right|$ equal to ?