Define for positive integer n, a function fn(x)=xnlnx (x>0). In the coordinate plane, denote by Sn the area of the figure enclosed by y=fn(x) (x≤t), the x-axis and the line x=t and denote by Tn the area of the rectagle with four vertices (1, 0), (t, 0), (t, fn(t)) and (1, fn(t)).(1) Find the local maximum fn(x).(2) When t moves in the range of t>1, find the value of t for which Tn(t)−Sn(t) is maximized.(3) Find S1(t) and Sn(t) (n≥2).(4) For each n≥2, prove that there exists the only t>1 such that Tn(t)=Sn(t).Note that you may use limx→∞xlnx=0. calculusintegrationfunctionlogarithmsanalytic geometrygeometrylimit