I can check ifa given function is the solution to a differential equation. I can translate a written statement into a differential equation. lean solve separable differential equations. 1. Consider the initial value problem ((1) (f) (g) (h) (i) dy _ 3x2+4x+2 121 h =0. dx 2(y1} , 3 wenx What is the sign of g at the point (0, 1), the point that we require to be on the solution curve? Show how you know. Is the numerator of the right hand side always positive, or only sometimes positive? Can the numerator ever be zero? Explain how you know. Keeping in mind your answer to part (b), what can you say about the slope of the solution curve to this initial value problem at points that have a y-coordinate close to, but less than, 1? Your explanation must make reference to your answer to part (b). Unjustied answers here will lead to unacceptable work. Based on your observations in parts (a}-(c], can the sign of % change from its value at (0,1)? Explain how you know. Keep in mind that solutions to differential equations must be continuous. Show by the method of separation of variables (not by substituting into the differential equation} that the implicit general solution to this equation is y22y2x3+2x2+2x+C Find the particular implicit solution that satises the given initial condition, that y = 1 when x = 0. Complete the square on the left, and use the result to show that this gives yzlim The result above gives two possible choices. Choose the one that gives the explicit solution to the initial value problem given at the start of this Learning Target. If you made a mistake and made the other choice in the previous part, what initial initial condi- tion is satised instead of the condition that y = 1 when x = 0 that is given in the