ClearAll; hour[k_] = Table[24 k + i - 25,{i,25}];
tide1 = {3.9, 3.4, 3.0, 2.8, 3.0, 3.6,
         4.4, 5.1, 5.7, 6.0, 5.6, 4.8,
         3.9, 3.4, 2.6, 1.9, 1.2, 1.0,
         1.3, 2.3, 3.2, 4.0, 4.3, 4.5,
         4.2};
data = Map[Thread,Transpose[{hour[1],tide1}]];
lp:= ListPlot[data]; interp = Interpolation[data];
plot:= Plot[interp[t],{t,0,24},PlotPoints->100];
Show[lp,plot,AxesOrigin->{-1,-0.25},
	 AxesLabel->{"t","y"}, PlotLabel->"y = f(t)"]
FindMinimum[interp[t],{t,17}]
FindMinimum[-interp[t],{t,9}]