Answer:
One real root, with multiplicity 7.
Step-by-step explanation:
The given expresion is

This is a seventh degree polynomial.
According to the fundamental theorem of Algebra, the expression has a potential 7 roots both real and complex.
This expression has one real root, with multiplicity , 7.
1. The solution for the equation is -16t² + 64t > 0
2. The interval notation for the equation is (0, 4)
Given,
The height of the building = 400 feet
A ball is thrown straight up from the top of the building.
The initial velocity of the ball = 64 feet/second
The height of the object is modeled by, s(t) = -16t² + 64t + 400
1. Solution for the equation:
s(t) > 400
So,
-16t² + 64t + 400 > 400
Add -400 to both sides
ie, -16t² + 64t + 400 - 400 > 400 - 400
We get,
-16t² + 64t > 0
2. Now find the interval notation for the equation:
-16t² + 64t > 0
Here, 64/16 = 4
So,
-16t (t - 4) > 0
Now,
t = 0 and t - 4 = 0
The interval notation for the equation is (0, 4)
Learn more about interval notation here:
brainly.com/question/28743524
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Answer:

Step-by-step explanation:
The derivative of
is calculated as follows:
![\dfrac{dF(x)}{dx}=\dfrac{d}{dx} [(5e^{4x})+(e^{-x^6})]](https://tex.z-dn.net/?f=%5Cdfrac%7BdF%28x%29%7D%7Bdx%7D%3D%5Cdfrac%7Bd%7D%7Bdx%7D%20%5B%285e%5E%7B4x%7D%29%2B%28e%5E%7B-x%5E6%7D%29%5D)
![\dfrac{dF(x)}{dx}=\dfrac{d}{dx} [(5e^{4x})]+\dfrac{d}{dx} [(e^{-x^6})]](https://tex.z-dn.net/?f=%5Cdfrac%7BdF%28x%29%7D%7Bdx%7D%3D%5Cdfrac%7Bd%7D%7Bdx%7D%20%5B%285e%5E%7B4x%7D%29%5D%2B%5Cdfrac%7Bd%7D%7Bdx%7D%20%5B%28e%5E%7B-x%5E6%7D%29%5D)
![\dfrac{dF(x)}{dx}=5\dfrac{d}{dx} [(e^{4x})]+\dfrac{d}{dx} [(e^{-x^6})]](https://tex.z-dn.net/?f=%5Cdfrac%7BdF%28x%29%7D%7Bdx%7D%3D5%5Cdfrac%7Bd%7D%7Bdx%7D%20%5B%28e%5E%7B4x%7D%29%5D%2B%5Cdfrac%7Bd%7D%7Bdx%7D%20%5B%28e%5E%7B-x%5E6%7D%29%5D)
using the chain rule we find that
![\dfrac{d}{dx} [(e^{4x})]= \dfrac{d}{d(4x)} [(e^{4x})]+ \dfrac{d}{dx} [4x] = 4e^{4x},](https://tex.z-dn.net/?f=%5Cdfrac%7Bd%7D%7Bdx%7D%20%5B%28e%5E%7B4x%7D%29%5D%3D%20%5Cdfrac%7Bd%7D%7Bd%284x%29%7D%20%5B%28e%5E%7B4x%7D%29%5D%2B%20%5Cdfrac%7Bd%7D%7Bdx%7D%20%5B4x%5D%20%3D%204e%5E%7B4x%7D%2C)
![\dfrac{d}{dx} [(e^{-x^6})] = \dfrac{d}{d(-x^6)} [(e^{-x^6})]+\dfrac{d}{dx} [(-x^6})]= -6x^5e^{-x^6};](https://tex.z-dn.net/?f=%5Cdfrac%7Bd%7D%7Bdx%7D%20%5B%28e%5E%7B-x%5E6%7D%29%5D%20%3D%20%5Cdfrac%7Bd%7D%7Bd%28-x%5E6%29%7D%20%5B%28e%5E%7B-x%5E6%7D%29%5D%2B%5Cdfrac%7Bd%7D%7Bdx%7D%20%5B%28-x%5E6%7D%29%5D%3D%20-6x%5E5e%5E%7B-x%5E6%7D%3B)
therefore,
![\dfrac{dF(x)}{dx}=5\dfrac{d}{dx} [(e^{4x})]+\dfrac{d}{dx} [(e^{-x^6})] =5(4e^{4x})-6x^5e^{x^{-6}}](https://tex.z-dn.net/?f=%5Cdfrac%7BdF%28x%29%7D%7Bdx%7D%3D5%5Cdfrac%7Bd%7D%7Bdx%7D%20%5B%28e%5E%7B4x%7D%29%5D%2B%5Cdfrac%7Bd%7D%7Bdx%7D%20%5B%28e%5E%7B-x%5E6%7D%29%5D%20%3D5%284e%5E%7B4x%7D%29-6x%5E5e%5E%7Bx%5E%7B-6%7D%7D)
