Writing:
points = {{5.667680905429296, 4.978997738176498},
{6.062672641119444, 4.821312039084925},
{6.463307024854181, 4.616923216296398},
{6.869389071217502, 4.358024801663491},
{7.280737790273085, 4.0328205724875374},
{7.697184816760531, 3.6220668338026347},
{8.118573184417112, 3.090204786097973},
{8.544756235833926, 2.354492374103027},
{8.97559665513716, 1.045159976700232},
{9.168625618559851, 6.355895552341976^-16},
{9.291801857401657, 6.34776334121662^-20}};
f = Interpolation[points];
Plot[f[x], {x, points[[1, 1]], points[[11, 1]]},
Epilog -> Point[points],
AxesLabel -> {x, y}]
I get:
which is what you want.