Answer:
![\frac{dy}{dx} =\frac{-8}{x^2} +2](https://tex.z-dn.net/?f=%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%5Cfrac%7B-8%7D%7Bx%5E2%7D%20%2B2)
![\frac{d^2y}{dx^2} =\frac{16}{x^3}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%5E2y%7D%7Bdx%5E2%7D%20%3D%5Cfrac%7B16%7D%7Bx%5E3%7D)
Stationary Points: See below.
General Formulas and Concepts:
<u>Pre-Algebra</u>
<u>Calculus</u>
Derivative Notation dy/dx
Derivative of a Constant equals 0.
Stationary Points are where the derivative is equal to 0.
- 1st Derivative Test - Tells us if the function f(x) has relative max or mins. Critical Numbers occur when f'(x) = 0 or f'(x) = undef
- 2nd Derivative Test - Tells us the function f(x)'s concavity behavior. Possible Points of Inflection/Points of Inflection occur when f"(x) = 0 or f"(x) = undef
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Quotient Rule: ![\frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%20%5B%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D%20%5D%3D%5Cfrac%7Bg%28x%29f%27%28x%29-g%27%28x%29f%28x%29%7D%7Bg%5E2%28x%29%7D)
Step-by-step explanation:
<u>Step 1: Define</u>
![f(x)=\frac{8}{x} +2x](https://tex.z-dn.net/?f=f%28x%29%3D%5Cfrac%7B8%7D%7Bx%7D%20%2B2x)
<u>Step 2: Find 1st Derivative (dy/dx)</u>
- Quotient Rule [Basic Power]:
![f'(x)=\frac{0(x)-1(8)}{x^2} +2x](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cfrac%7B0%28x%29-1%288%29%7D%7Bx%5E2%7D%20%2B2x)
- Simplify:
![f'(x)=\frac{-8}{x^2} +2x](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cfrac%7B-8%7D%7Bx%5E2%7D%20%2B2x)
- Basic Power Rule:
![f'(x)=\frac{-8}{x^2} +1 \cdot 2x^{1-1}](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cfrac%7B-8%7D%7Bx%5E2%7D%20%2B1%20%5Ccdot%202x%5E%7B1-1%7D)
- Simplify:
![f'(x)=\frac{-8}{x^2} +2](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cfrac%7B-8%7D%7Bx%5E2%7D%20%2B2)
<u>Step 3: 1st Derivative Test</u>
- Set 1st Derivative equal to 0:
![0=\frac{-8}{x^2} +2](https://tex.z-dn.net/?f=0%3D%5Cfrac%7B-8%7D%7Bx%5E2%7D%20%2B2)
- Subtract 2 on both sides:
![-2=\frac{-8}{x^2}](https://tex.z-dn.net/?f=-2%3D%5Cfrac%7B-8%7D%7Bx%5E2%7D)
- Multiply x² on both sides:
![-2x^2=-8](https://tex.z-dn.net/?f=-2x%5E2%3D-8)
- Divide -2 on both sides:
![x^2=4](https://tex.z-dn.net/?f=x%5E2%3D4)
- Square root both sides:
![x= \pm 2](https://tex.z-dn.net/?f=x%3D%20%5Cpm%202)
Our Critical Points (stationary points for rel max/min) are -2 and 2.
<u>Step 4: Find 2nd Derivative (d²y/dx²)</u>
- Define:
![f'(x)=\frac{-8}{x^2} +2](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cfrac%7B-8%7D%7Bx%5E2%7D%20%2B2)
- Quotient Rule [Basic Power]:
![f''(x)=\frac{0(x^2)-2x(-8)}{(x^2)^2} +2](https://tex.z-dn.net/?f=f%27%27%28x%29%3D%5Cfrac%7B0%28x%5E2%29-2x%28-8%29%7D%7B%28x%5E2%29%5E2%7D%20%2B2)
- Simplify:
![f''(x)=\frac{16}{x^3} +2](https://tex.z-dn.net/?f=f%27%27%28x%29%3D%5Cfrac%7B16%7D%7Bx%5E3%7D%20%2B2)
- Basic Power Rule:
![f''(x)=\frac{16}{x^3}](https://tex.z-dn.net/?f=f%27%27%28x%29%3D%5Cfrac%7B16%7D%7Bx%5E3%7D)
<u>Step 5: 2nd Derivative Test</u>
- Set 2nd Derivative equal to 0:
![0=\frac{16}{x^3}](https://tex.z-dn.net/?f=0%3D%5Cfrac%7B16%7D%7Bx%5E3%7D)
- Solve for <em>x</em>:
![x = 0](https://tex.z-dn.net/?f=x%20%3D%200)
Our Possible Point of Inflection (stationary points for concavity) is 0.
<u>Step 6: Find coordinates</u>
<em>Plug in the C.N and P.P.I into f(x) to find coordinate points.</em>
x = -2
- Substitute:
![f(-2)=\frac{8}{-2} +2(-2)](https://tex.z-dn.net/?f=f%28-2%29%3D%5Cfrac%7B8%7D%7B-2%7D%20%2B2%28-2%29)
- Divide/Multiply:
![f(-2)=-4-4](https://tex.z-dn.net/?f=f%28-2%29%3D-4-4)
- Subtract:
![f(-2)=-8](https://tex.z-dn.net/?f=f%28-2%29%3D-8)
x = 2
- Substitute:
![f(2)=\frac{8}{2} +2(2)](https://tex.z-dn.net/?f=f%282%29%3D%5Cfrac%7B8%7D%7B2%7D%20%2B2%282%29)
- Divide/Multiply:
![f(2)=4 +4](https://tex.z-dn.net/?f=f%282%29%3D4%20%2B4)
- Add:
![f(2)=8](https://tex.z-dn.net/?f=f%282%29%3D8)
x = 0
- Substitute:
![f(0)=\frac{8}{0} +2(0)](https://tex.z-dn.net/?f=f%280%29%3D%5Cfrac%7B8%7D%7B0%7D%20%2B2%280%29)
- Evaluate:
![f(0)=\text{unde} \text{fined}](https://tex.z-dn.net/?f=f%280%29%3D%5Ctext%7Bunde%7D%20%5Ctext%7Bfined%7D)
<u>Step 7: Identify Behavior</u>
<em>See Attachment.</em>
Point (-2, -8) is a relative max because f'(x) changes signs from + to -.
Point (2, 8) is a relative min because f'(x) changes signs from - to +.
When x = 0, there is a concavity change because f"(x) changes signs from - to +.