Answer:
Stationary Points: See below.
General Formulas and Concepts:
<u>Pre-Algebra</u>
- Equality Properties
<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:
Step-by-step explanation:
<u>Step 1: Define</u>
<u>Step 2: Find 1st Derivative (dy/dx)</u>
- Quotient Rule [Basic Power]:
- Simplify:
- Basic Power Rule:
- Simplify:
<u>Step 3: 1st Derivative Test</u>
- Set 1st Derivative equal to 0:
- Subtract 2 on both sides:
- Multiply x² on both sides:
- Divide -2 on both sides:
- Square root both sides:
Our Critical Points (stationary points for rel max/min) are -2 and 2.
<u>Step 4: Find 2nd Derivative (d²y/dx²)</u>
- Define:
- Quotient Rule [Basic Power]:
- Simplify:
- Basic Power Rule:
<u>Step 5: 2nd Derivative Test</u>
- Set 2nd Derivative equal to 0:
- Solve for <em>x</em>:
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:
- Divide/Multiply:
- Subtract:
x = 2
- Substitute:
- Divide/Multiply:
- Add:
x = 0
- Substitute:
- Evaluate:
<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 +.
