We are all equal but some are more equal than others

Each cake has to be cut by at least two cuts. Do that, and you have used up 6 cuts to make 12 pieces.

To get the remaining pieces with as few cuts as possible, do all the rest of the cutting on the same cake, so that as many lines as possible can be crossed with every cut. In fact, each cut should cross all the lines existing before that cut. Each time $n$ lines are crossed, $n+1$ pieces are cut, each of them becoming two pieces, so $n+1$ pieces are generated.

Starting with the two cuts already done, we will cross those two cuts making 3 new pieces. Then cross 3 cuts making 4 pieces. Next we'll make 5, 6, and finally we could do 7, but we can do fewer, to limit ourselves to the 34. It's always possible to avoid crossing a few cuts and manufacture fewer pieces per cut.

When we have accumulated enough pieces, we just count up the cuts made, and it in this case it comes to a total of 11.

This is a classic case of Circle Division by lines .

In this case, we have $N(3) + N(4) + N(5) = 34$ thus the upper bound is $3+4+5=12$ .

However, as the Challenge Master has pointed out, because $f(x)$ is a quadratic that concave ups, we can apply Jensen's inequality. so then extremal principle is applied. Then we want to set $f(a) + f(b) + f(c) \ge 34$ .

WLOG $a\leq b \leq c$ , we get $a=b=2,c=7$ . thus the sum is minimized with $a+b+c=11$ .