Twice Differentiable Function Intersection

Let f:[0,1]→R be a twice differentiable function in (0,1) such that f(0)−3 and f(1)−5. If the line y−2x+3 intersects the graph of f at only two distinct points in (0,1), then the least number of points x∈(0,1), at which f′′(x)−0, is _______.