题目

Jim Denevan is an amazing artist whose work is admired by all, but owned by none--that’s because all of Jim’s art is created on an unusual canvas（画布）--the soft sand. He sometimes spends days working on a piece, only to see it washed away by the sea or a storm and that is just the way he likes it.Denevan discovered his artistic talents about ten years ago, when he was wandering aimlessly on the beach with a stick. He ended up drawing a 12-foot-long fish. Since then, Jim has traveled over 1,800 miles while creating over 600 pieces of sand art.Over the years, his drawings have become 设函数f（x）=-x3+ax2+a2x+1（x∈R），其中a∈R．（Ⅰ）当a=1时，求曲线y=f（x）在点（2，f（2））处的切线方程；（Ⅱ）当a≠0时，求函数f（x）的极大值和极小值；（Ⅲ）当a=2时，是否存在函数y=f（x）图象上两点以及函数y=f'（x）图象上两点，使得以这四点为顶点的四边形ABCD满足如下条件：①四边形ABCD是平行四边形；②AB⊥x轴；③|AB|=4．若存在，指出四边形ABCD的个数；若不存在，说明理由．