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: ![\frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%20%5B%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D%20%5D%3D%5Cfrac%7Bg%28x%29f%27%28x%29-g%27%28x%29f%28x%29%7D%7Bg%5E2%28x%29%7D)
Chain Rule: ![\frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdx%7D%5Bf%28g%28x%29%29%5D%20%3Df%27%28g%28x%29%29%20%5Ccdot%20g%27%28x%29)
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: 