How to find local minimum of a interpolated function?

Questions
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1

I’m not sure what went wrong but I can’t seem to output the local minimum x values of the interpolated function below.

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2

You are using FindMinimum with wrong syntax. I would rather use Minimize:

Minimize[{q[x], 1 <= x <= 88}, x]

but it gives a wrong answer:

Min[n]
Position[n, %]