In the plane rectangular coordinate system xoy, it is known that P is the moving point on the image of the function f (x) = ex (x > 0). The tangent l of the image at point P intersects the y-axis at point m, and the vertical line passing through point P intersects the y-axis at point n. suppose the ordinate of the midpoint of line Mn is t, then the maximum value of T is______ ．

In the plane rectangular coordinate system xoy, it is known that P is the moving point on the image of the function f (x) = ex (x > 0). The tangent l of the image at point P intersects the y-axis at point m, and the vertical line passing through point P intersects the y-axis at point n. suppose the ordinate of the midpoint of line Mn is t, then the maximum value of T is______ ．

Let the coordinate of tangent point be (m, EM) ｜ the equation of tangent line L of the image at point p be y-em = EM (x-m) let x = 0, the equation of tangent line where y = (1-m) em passes through point P as l is y-em = - E-M (x-m) let x = 0, the equation of y = EM + me-m ｜ the ordinate of the midpoint of line Mn is t = 12 [(2-m) em + me-m] t '= 12 [- em + (2-m) em + e-m-me-m], let t' = 0, the solution is m = 1, when m ∈ (0, 1), t '> 0, when m ∈ (0, 1), t' > 0 When m = 1, the maximum value of T is 12 (E + e − 1), so the answer is: 12 (E + e − 1)