Answer:
Concave Up Interval:
Concave Down Interval:
General Formulas and Concepts:
<u>Calculus</u>
Derivative of a Constant is 0.
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Quotient Rule:
Chain Rule:
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
Step-by-step explanation:
<u>Step 1: Define</u>
<u>Step 2: Find 2nd Derivative</u>
- 1st Derivative [Quotient/Chain/Basic]:
- Simplify 1st Derivative:
- 2nd Derivative [Quotient/Chain/Basic]:
- Simplify 2nd Derivative:
<u>Step 3: Find P.P.I</u>
- Set f"(x) equal to zero:
<em>Case 1: f" is 0</em>
- Solve Numerator:
- Divide 6:
- Add 1:
- Divide 3:
- Square root:
- Simplify:
- Rewrite:
<em>Case 2: f" is undefined</em>
- Solve Denominator:
- Cube root:
- Subtract 1:
We don't go into imaginary numbers when dealing with the 2nd Derivative Test, so our P.P.I is (x ≈ ±0.57735).
<u>Step 4: Number Line Test</u>
<em>See Attachment.</em>
We plug in the test points into the 2nd Derivative and see if the P.P.I is a P.I.
x = -1
- Substitute:
- Exponents:
- Multiply:
- Subtract/Add:
- Exponents:
- Multiply:
- Simplify:
This means that the graph f(x) is concave up before .
x = 0
- Substitute:
- Exponents:
- Multiply:
- Subtract/Add:
- Exponents:
- Multiply:
- Divide:
This means that the graph f(x) is concave down between and .
x = 1
- Substitute:
- Exponents:
- Multiply:
- Subtract/Add:
- Exponents:
- Multiply:
- Simplify:
This means that the graph f(x) is concave up after .
<u>Step 5: Identify</u>
Since f"(x) changes concavity from positive to negative at and changes from negative to positive at
, then we know that the P.P.I's
are actually P.I's.
Let's find what actual <em>point </em>on f(x) when the concavity changes.
- Substitute in P.I into f(x):
- Evaluate Exponents:
- Add:
- Divide:
- Substitute in P.I into f(x):
- Evaluate Exponents:
- Add:
- Divide:
<u>Step 6: Define Intervals</u>
We know that <em>before </em>f(x) reaches , the graph is concave up. We used the 2nd Derivative Test to confirm this.
We know that <em>after </em>f(x) passes , the graph is concave up. We used the 2nd Derivative Test to confirm this.
Concave Up Interval:
We know that <em>after</em> f(x) <em>passes</em> , the graph is concave up <em>until</em>
. We used the 2nd Derivative Test to confirm this.
Concave Down Interval:
