Question

Can anybody help plzz?? 65 points

Answer 1

`(dy)/(dx) =(-8)/(x^2) +2`

`(d^2y)/(dx^2) =(16)/(x^3)`

Stationary Points: See below.

General Formulas and Concepts:

Pre-Algebra

• Equality Properties

Calculus

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:
`(d)/(dx) [(f(x))/(g(x)) ]=(g(x)f'(x)-g'(x)f(x))/(g^2(x))`

Explanation:

Step 1: Define

`f(x)=(8)/(x) +2x`

Step 2: Find 1st Derivative (dy/dx)

1. Quotient Rule [Basic Power]:
   `f'(x)=(0(x)-1(8))/(x^2) +2x`
2. Simplify:
   `f'(x)=(-8)/(x^2) +2x`
3. Basic Power Rule:
   `f'(x)=(-8)/(x^2) +1 \cdot 2x^(1-1)`
4. Simplify:
   `f'(x)=(-8)/(x^2) +2`

Step 3: 1st Derivative Test

1. Set 1st Derivative equal to 0:
   `0=(-8)/(x^2) +2`
2. Subtract 2 on both sides:
   `-2=(-8)/(x^2)`
3. Multiply x² on both sides:
   `-2x^2=-8`
4. Divide -2 on both sides:
   `x^2=4`
5. Square root both sides:
   `x= \pm 2`

Our Critical Points (stationary points for rel max/min) are -2 and 2.

Step 4: Find 2nd Derivative (d²y/dx²)

1. Define:
   `f'(x)=(-8)/(x^2) +2`
2. Quotient Rule [Basic Power]:
   `f''(x)=(0(x^2)-2x(-8))/((x^2)^2) +2`
3. Simplify:
   `f''(x)=(16)/(x^3) +2`
4. Basic Power Rule:
   `f''(x)=(16)/(x^3)`

Step 5: 2nd Derivative Test

1. Set 2nd Derivative equal to 0:
   `0=(16)/(x^3)`
2. Solve for x:
   `x = 0`

Our Possible Point of Inflection (stationary points for concavity) is 0.

Step 6: Find coordinates

Plug in the C.N and P.P.I into f(x) to find coordinate points.

x = -2

1. Substitute:
   `f(-2)=(8)/(-2) +2(-2)`
2. Divide/Multiply:
   `f(-2)=-4-4`
3. Subtract:
   `f(-2)=-8`

x = 2

1. Substitute:
   `f(2)=(8)/(2) +2(2)`
2. Divide/Multiply:
   `f(2)=4 +4`
3. Add:
   `f(2)=8`

x = 0

1. Substitute:
   `f(0)=(8)/(0) +2(0)`
2. Evaluate:
   `f(0)=\text{unde} \text{fined}`

Step 7: Identify Behavior

See Attachment.

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 +.