Answer:
Concave Up: (-∞, 1)
Concave Down: (1, ∞)
General Formulas and Concepts:
<u>Pre-Algebra</u>
- Order of Operations: BPEMDAS
- Equality Properties
<u>Algebra I</u>
<u>Calculus</u>
Derivative of a Constant is 0.
Basic Power Rule:
Second Derivative Test:
- Possible Points of Inflection (P.P.I) - Tells us the <em>possible</em> x-values where the graph f(x) may change concavity. Occurs when f"(x) = 0 or undefined
- Points of Inflection (P.I) - Actual x-values when the graph f(x) changes concavity
- Number Line Test - Helps us determine whether a P.P.I is a P.I
Step-by-step explanation:
<u>Step 1: Define</u>
f(x) = -x³ + 3x² - 2
<u>Step 2: Find 2nd Derivative</u>
- 1st Derivative [Basic Power]: f'(x) = -1 · 3x³⁻² + 2 · 3x²⁻¹
- Simplify: f'(x) = -3x² + 6x
- 2nd Derivative [Basic Power]: f"(x) = -3 · 2x²⁻¹ + 1 · 6x¹⁻¹
- Simplify: f"(x) = -6x + 6
<u>Step 3: Find P.P.I</u>
- Set f"(x) equal to zero: 0 = -6x + 6
- Isolate <em>x</em> term: -6 = -6x
- Isolate <em>x</em>: 1 = x
- Rewrite: P.P.I x = 1
<u>Step 4: Number Line Test</u>
<em>See Attachment.</em>
We plug in the test points <em>into</em> the 2nd Derivative and see if the P.P.I is a P.I.
x = 0
- Substitute: f"(0) = -6(0) + 6
- Multiply: f"(0) = 0 + 6
- Add: f"(0) = 6
This means that the graph f(x) is concave up before x = 1.
x = 2
- Substitute: f"(2) = -6(2) + 6
- Multiply: f"(2) = -12 + 6
- Add: f"(2) = -6
This means that the graph f(x) is concave down after x = 1.
<u>Step 5: Identify</u>
Since f"(x) changes concavity from positive to negative at x = 1, then we know that the P.P.I x = 1 is actually a P.I x = 1.
Let's find what actual <em>point</em> on f(x) when the concavity changes.
- Substitute in P.I into f(x): f(1) = -(1)³ + 3(1)² - 2
- Evaluate Exponents: f(1) = -(1) + 3(1) - 2
- Multiply: f(1) = -1 + 3 - 2
- Combine like terms: f(1) = 0
So at (1, 0), f(x) changes concavity from concave up to concave down.
<u>Step 6: Define Intervals</u>
We know that <em>before</em> f(x) reaches x = 1, the graph is <em>concave up</em>. We used the 2nd Derivative Test to confirm this.
Concave Up Interval: (-∞, 1)
We also know that <em>after </em>f(x) passes x = 1, the graph is <em>concave down.</em> We used the 2nd Derivative Test to confirm this as well.
Concave Down Interval: (1, ∞)