Converge to zero always?

Hi,
Is the lowest point of the cost function in the contour plot always zero?
I have doubts

No. It is just a function, and most functions have minimums that are not zero. Just that we expect the cost function minimum to be a positive low number if it is not zero, since our model may not fit the data exactly.

The optimum solution is where the cost is minimized.
It does not have to be a small number.

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