Answer:
(27.55, 7.22), (-11.3, 3.21).
Step-by-step explanation:
When is the tangent to the curve horizontal?
The tangent curve is horizontal when the derivative is zero.
The derivative is:
![\frac{dy}{dx} = 3t^{2} - 2t - 27](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%203t%5E%7B2%7D%20-%202t%20-%2027)
Solving a quadratic equation:
Given a second order polynomial expressed by the following equation:
.
This polynomial has roots
such that
, given by the following formulas:
![x_{1} = \frac{-b + \sqrt{\bigtriangleup}}{2*a}](https://tex.z-dn.net/?f=x_%7B1%7D%20%3D%20%5Cfrac%7B-b%20%2B%20%5Csqrt%7B%5Cbigtriangleup%7D%7D%7B2%2Aa%7D)
![x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a}](https://tex.z-dn.net/?f=x_%7B2%7D%20%3D%20%5Cfrac%7B-b%20-%20%5Csqrt%7B%5Cbigtriangleup%7D%7D%7B2%2Aa%7D)
![\bigtriangleup = b^{2} - 4ac](https://tex.z-dn.net/?f=%5Cbigtriangleup%20%3D%20b%5E%7B2%7D%20-%204ac)
In this question:
![3t^{2} - 2t - 27 = 0](https://tex.z-dn.net/?f=3t%5E%7B2%7D%20-%202t%20-%2027%20%3D%200)
So
![a = 3, b = -2, c = -27](https://tex.z-dn.net/?f=a%20%3D%203%2C%20b%20%3D%20-2%2C%20c%20%3D%20-27)
Then
![\bigtriangleup = b^{2} - 4ac = (-2)^{2} - 4*3*(-27) = 328](https://tex.z-dn.net/?f=%5Cbigtriangleup%20%3D%20b%5E%7B2%7D%20-%204ac%20%3D%20%28-2%29%5E%7B2%7D%20-%204%2A3%2A%28-27%29%20%3D%20328)
So
![t_{1} = \frac{-(-2) + \sqrt{328}}{2*3} = 3.35](https://tex.z-dn.net/?f=t_%7B1%7D%20%3D%20%5Cfrac%7B-%28-2%29%20%2B%20%5Csqrt%7B328%7D%7D%7B2%2A3%7D%20%3D%203.35)
![t_{2} = \frac{-(-2) - \sqrt{328}}{2*3} = -2.685](https://tex.z-dn.net/?f=t_%7B2%7D%20%3D%20%5Cfrac%7B-%28-2%29%20-%20%5Csqrt%7B328%7D%7D%7B2%2A3%7D%20%3D%20-2.685)
Enter your answers as a comma-separated list of ordered pairs.
We found values of t, now we have to replace in the equations for x and y.
t = 3.35
![x = t^{3} - 3t = (3.35)^{3} - 3*3.35 = 27.55](https://tex.z-dn.net/?f=x%20%3D%20t%5E%7B3%7D%20-%203t%20%3D%20%283.35%29%5E%7B3%7D%20-%203%2A3.35%20%3D%2027.55)
![y = t^{2} - 4 = (3.35)^2 - 4 = 7.22](https://tex.z-dn.net/?f=y%20%3D%20t%5E%7B2%7D%20-%204%20%3D%20%283.35%29%5E2%20-%204%20%3D%207.22)
The first point is (27.55, 7.22)
t = -2.685
![x = t^{3} - 3t = (-2.685)^3 - 3*(-2.685) = -11.3](https://tex.z-dn.net/?f=x%20%3D%20t%5E%7B3%7D%20-%203t%20%3D%20%28-2.685%29%5E3%20-%203%2A%28-2.685%29%20%3D%20-11.3)
![y = t^{2} - 4 = (-2.685)^2 - 4 = 3.21](https://tex.z-dn.net/?f=y%20%3D%20t%5E%7B2%7D%20-%204%20%3D%20%28-2.685%29%5E2%20-%204%20%3D%203.21)
The second point is (-11.3, 3.21).