ClearAll; hour[k_] = Table[24 k + i - 25,{i,25}];
tide3 = {4.6, 4.2, 3.5, 3.0, 2.6, 
         2.5, 2.8, 3.5, 4.3, 4.9, 
         5.4, 5.5, 5.1, 4.4, 3.5, 
         2.8, 2.2, 1.7, 1.5, 1.8, 
         2.6, 3.4, 4.1, 4.5, 4.5};
data = Map[Thread,Transpose[{hour[3],tide3}]];
lp:= ListPlot[data]; interp = Interpolation[data];
plot:= Plot[interp[t],{t,48,72},PlotPoints->100];
Show[lp,plot,AxesOrigin->{47,1.25},
	 AxesLabel->{"t","y"}, PlotLabel->"y = f(t)"]
FindMinimum[interp[t],{t,64}]
FindMinimum[-interp[t],{t,57}]