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HallsofIvy
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As long as I have been working problems like this, and teaching Linear Algebra (one of my favorite courses) I still consider writing a system of equations as a matrix and row reducing to be more trouble than it is worth!
A general cubic is ax^3+ bx^2+ cx+ d. The fact that the point (2, 0) is on graph means that 8a+ 4b+ 2c+ d= 0. The fact that (-2, 6) is on the graph means that -8a+ 4b- 2c+ d= 6. The fact that there is a horizontal tangent at x= 2 means that 12a+ 4b+ c= 0. The fact that there is a horizontal tangent at x= -2 means that 12a- 4b+ c= 0. We need to solve those 4 equations for a, b, c, and d.
Adding 8a+ 4b+ 2c+ d= 0 and -8a+ 4b-2c+ d= 6 gives 8b+ 2d= 6 or d= 3- 4b. Adding 12a+ 4b+ c= 0 and 12a- 4b+ c= 0 gives 24a+ 2c= 0 or c= -12a. Putting those into the first and third equations gives 8a+ 4b+ 2(-12a)+ 3-4b= -16a+ 3= 0 or a= 3/16 and 12a+ 4b+ (-12a)= 4b= 0. Then c= -12a= -36/16= -9/4 and d= 3.
A general cubic is ax^3+ bx^2+ cx+ d. The fact that the point (2, 0) is on graph means that 8a+ 4b+ 2c+ d= 0. The fact that (-2, 6) is on the graph means that -8a+ 4b- 2c+ d= 6. The fact that there is a horizontal tangent at x= 2 means that 12a+ 4b+ c= 0. The fact that there is a horizontal tangent at x= -2 means that 12a- 4b+ c= 0. We need to solve those 4 equations for a, b, c, and d.
Adding 8a+ 4b+ 2c+ d= 0 and -8a+ 4b-2c+ d= 6 gives 8b+ 2d= 6 or d= 3- 4b. Adding 12a+ 4b+ c= 0 and 12a- 4b+ c= 0 gives 24a+ 2c= 0 or c= -12a. Putting those into the first and third equations gives 8a+ 4b+ 2(-12a)+ 3-4b= -16a+ 3= 0 or a= 3/16 and 12a+ 4b+ (-12a)= 4b= 0. Then c= -12a= -36/16= -9/4 and d= 3.