Function intersecting 3 points & deriviate is positive for a range of x values

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Thank you for taking the time to help out on this question.

I'm looking for a function that intersects 3 points, and a derivative for every value of x between x=0 and x = 365 where dy/dx >= 0.

My points are:

Point 1: (0,0) or [(1,1) if needed]

Point 2: (x2,y2) where 0 < x < 365 and 1 < y2 < MaxValue (MaxValue is given)

Point 3: (365, MaxValue)

I've first setup a system of equations using explicitly ax^2 + bx + c, then I tried aln(bx) + c, and then ae^bx + c, etc but the values of a,b,c fail the second test of needing the derivative to be greater than or equal to zero. Any thoughts or suggestions would be immensely helpful.