) that has horizontal tangents at the points (-2,2) and (4,-4).
= ...............
Marion J. answered • 02/18/22
Qualified teacher who loves all types of Math
Horizontal tangents have a slope of zero, so put the derivative of your function equal to zero for the two cases given: x = -2 and x = 4:
f'(x) = 3ax2 + 2bx + c , so 3a(-2)2 + 2b(-2) + c = 0 and 3a(4)2 + 2b(4) + c = 0
These simplify to: 12a - 4b + c = 0 and 48a + 8b + c = 0
We can also create two equations using the points (-2,2) and (4,-4):
2 = a(-2)3 + b(-2)2 + c(-2) + d which simplifies to: 2 = -8a +4b -2c + d
-4 = a(4)3 + b(4)2 + c(4) + d which simplifies to: -4 = 64a + 16b + 4c + d
You now have a system of 4 equations to solve for the 4 unknowns: a, b, c, and d:
12a - 4b + c + 0d = 0
48a + 8b + c + 0d = 0
-8a + 4b - 2c + d = 2
64a + 16b + 4c + d = -4
Create a 4 x 5 matrix and solve on your calculator.
This gives a = 1/18 b = -1/6 c = -4/3 d = 4/9
So f(x) = 1/18 x3 - 1/6 x2 - 4/3 x + 4/9
Mariam A.
I have a question how can we do the matrix 4x5 on the calculator? what are the steps?02/26/22
Marion J.
It will depend on which calculator you have. If your teacher hasn't shown you how to do it, I would recommend searching online for [your calculator model] and "solving systems of equations using matrices".02/26/22
Mariam A.
Thank you so much for your help !!02/18/22