Answer:
-15804
Step-by-step explanation:
Take the derivative:
g’(x) = 12x^3 - 24x^2
Set equal to zero and solve:
0 = 12x^3 - 24x^2
0 = 12x^2 (x - 2)
x = 0 or x = 2
Plug back into original
g(0) = 3(0^4) - 8(0^3)
g(0) = 0 - 0
g(0) = 0
g(2) = 3(2^2) - 8(2^3)
g(2) = 3(4) - 8(8)
g(2) = 12 - 64
g(2) = -52
There is an absolute max at (0,0) or when x = 0
<span>32 - 2x < 18 is the answer</span>
Lets simplify each one of the rational expressions using the product rule of exponents:
![a^n*a^m=a^{n+m}](https://tex.z-dn.net/?f=a%5En%2Aa%5Em%3Da%5E%7Bn%2Bm%7D)
and the quotient rule of exponents:
![\frac{a^n}{a^m} =a^{n-m}](https://tex.z-dn.net/?f=%20%5Cfrac%7Ba%5En%7D%7Ba%5Em%7D%20%3Da%5E%7Bn-m%7D)
.
1.
![\frac{-3x^2}{2y^2} * \frac{y^2}{9x} =- \frac{x}{6} = \frac{-1}{6} x](https://tex.z-dn.net/?f=%20%5Cfrac%7B-3x%5E2%7D%7B2y%5E2%7D%20%2A%20%5Cfrac%7By%5E2%7D%7B9x%7D%20%3D-%20%5Cfrac%7Bx%7D%7B6%7D%20%3D%20%5Cfrac%7B-1%7D%7B6%7D%20x)
2.
![\frac{5y^2}{10x^2} * \frac{4x^2}{y} =2y](https://tex.z-dn.net/?f=%20%5Cfrac%7B5y%5E2%7D%7B10x%5E2%7D%20%2A%20%5Cfrac%7B4x%5E2%7D%7By%7D%20%3D2y)
3.
![\frac{-9y^2}{5x} * \frac{-10x^2}{3y} =6xy](https://tex.z-dn.net/?f=%20%5Cfrac%7B-9y%5E2%7D%7B5x%7D%20%2A%20%5Cfrac%7B-10x%5E2%7D%7B3y%7D%20%3D6xy)
4.
We can conclude that you should pair the rational expressions with their simplest form as follows:
![6xy-----\ \textgreater \ \frac{-9y^2}{5x} * \frac{-10x^2}{3y}](https://tex.z-dn.net/?f=6xy-----%5C%20%5Ctextgreater%20%5C%20%5Cfrac%7B-9y%5E2%7D%7B5x%7D%20%2A%20%5Cfrac%7B-10x%5E2%7D%7B3y%7D)
![\frac{-1}{6} x-----\ \textgreater \ \frac{-3x^2}{2y^2} * \frac{y^2}{9x}](https://tex.z-dn.net/?f=%5Cfrac%7B-1%7D%7B6%7D%20x-----%5C%20%5Ctextgreater%20%5C%20%20%5Cfrac%7B-3x%5E2%7D%7B2y%5E2%7D%20%2A%20%5Cfrac%7By%5E2%7D%7B9x%7D%20)
![-4x-----\ \textgreater \ \frac{14x^2}{5y} * \frac{-10y}{7x}](https://tex.z-dn.net/?f=-4x-----%5C%20%5Ctextgreater%20%5C%20%5Cfrac%7B14x%5E2%7D%7B5y%7D%20%2A%20%5Cfrac%7B-10y%7D%7B7x%7D)