I dont know why the answer is this please explain Explain wh

I don\'t know why the answer is this. please explain.

Explain why the functions with the given graphs can\'t be solutions of the differential equation dy/dt = et(y - 1)2. This function is increasing and decreasing. But dy/dt = et(y - 1)20f or all t, implying that the graph of the solution of the differential equation cannot be decreasing on any interval. When y = 1, dy/dt = 0, but the graph does not have a horizontal tangent line. Need Help? Read it Chat About it

Solution

a) The graph represents the solution to the differential equation given, or in other words, the differential equation represents the first-derivative of the graph. The first derivative test tells us whether a function is increasing or decreasing on a particular interval. For instance, if the derivative is greater than 0 it is increasing and if it is less than 0 it is decreasing. Since the derivative (or the differential equation) is greater than 0 for ALL t, the solution must always be increasing. The graph given is sometimes increasing but at other times decreasing so it cannot be a solution. b) Again, the differential equation represents the first derivative of the graphed function. The derivative of a function represents the slope of a tangent line at a particular point. Since the derivative equals 0 at the point where y=1, the tangent line at y=1 should have 0 slope (be horizontal). Since a tangent line on the graph at y=1 would not have 0 slope, it cannot be a solution.