Question

The graph of f (x) = −x3 + 3x2 − 2 is

shown in the figure to the right.
Complete the table below by using
the second derivative of f to identify
the intervals on which f us concave
upward or concave downward and
to identify the inflection points of f.

Answer 1

Concave Up: (-∞, 1)

Concave Down: (1, ∞)

General Formulas and Concepts:

Pre-Algebra

• Order of Operations: BPEMDAS
• Equality Properties

Algebra I

• Combining Like Terms

Calculus

Derivative of a Constant is 0.

Basic Power Rule:

• f(x) = cxⁿ

• f’(x) = c·nxⁿ⁻¹

Second Derivative Test:

• Possible Points of Inflection (P.P.I) - Tells us the possible 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

Explanation:

Step 1: Define

f(x) = -x³ + 3x² - 2

Step 2: Find 2nd Derivative

1. 1st Derivative [Basic Power]: f'(x) = -1 · 3x³⁻² + 2 · 3x²⁻¹
2. Simplify: f'(x) = -3x² + 6x
3. 2nd Derivative [Basic Power]: f"(x) = -3 · 2x²⁻¹ + 1 · 6x¹⁻¹
4. Simplify: f"(x) = -6x + 6

Step 3: Find P.P.I

1. Set f"(x) equal to zero: 0 = -6x + 6
2. Isolate x term: -6 = -6x
3. Isolate x: 1 = x
4. Rewrite: P.P.I x = 1

Step 4: Number Line Test

See Attachment.

We plug in the test points into the 2nd Derivative and see if the P.P.I is a P.I.

x = 0

1. Substitute: f"(0) = -6(0) + 6
2. Multiply: f"(0) = 0 + 6
3. Add: f"(0) = 6

This means that the graph f(x) is concave up before x = 1.

x = 2

1. Substitute: f"(2) = -6(2) + 6
2. Multiply: f"(2) = -12 + 6
3. Add: f"(2) = -6

This means that the graph f(x) is concave down after x = 1.

Step 5: Identify

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 point on f(x) when the concavity changes.

1. Substitute in P.I into f(x): f(1) = -(1)³ + 3(1)² - 2
2. Evaluate Exponents: f(1) = -(1) + 3(1) - 2
3. Multiply: f(1) = -1 + 3 - 2
4. Combine like terms: f(1) = 0

So at (1, 0), f(x) changes concavity from concave up to concave down.

Step 6: Define Intervals

We know that before f(x) reaches x = 1, the graph is concave up. We used the 2nd Derivative Test to confirm this.

Concave Up Interval: (-∞, 1)

We also know that after f(x) passes x = 1, the graph is concave down. We used the 2nd Derivative Test to confirm this as well.

Concave Down Interval: (1, ∞)