ClearAll; hour[k_] = Table[24 k + i - 25,{i,25}];
tide4 = {4.5, 4.2, 3.7, 3.1, 2.5, 
         2.2, 2.2, 2.6, 3.3, 4.1, 
         4.6, 4.9, 4.8, 4.3, 3.6, 
         2.9, 2.2, 1.6, 1.3, 1.4, 
         2.0, 2.8, 3.6, 4.1, 4.4};
data = Map[Thread,Transpose[{hour[4],tide4}]];
lp:= ListPlot[data]; interp = Interpolation[data];
plot:= Plot[interp[t],{t,72,96},PlotPoints->100];
Show[lp,plot,AxesOrigin->{71,1.0},
	 AxesLabel->{"t","y"}, PlotLabel->"y = f(t)"]
FindMinimum[interp[t],{t,90}]
FindMinimum[-interp[t],{t,83}]