To analyze and solve the problem, let’s break down the provided function $f(x) = 4(x - 3)^6 + 2(x - 3)^4$ and determine its local extrema, concavity, and points of inflection.

Step 1: Find the Derivative for Extrema

To identify local extrema, we start by finding the first derivative, $f'(x)$.

1. $f(x) = 4(x - 3)^6 + 2(x - 3)^4$
2. Differentiate term-by-term:
   • For $4(x - 3)^6$, use the chain rule: $4 \cdot 6(x - 3)^5 = 24(x - 3)^5$.
   • For $2(x - 3)^4$, use the chain rule: $2 \cdot 4(x - 3)^3 = 8(x - 3)^3$.

So, $f'(x) = 24(x - 3)^5 + 8(x - 3)^3$ 3. Factor out $8(x - 3)^3$: $f'(x) = 8(x - 3)^3(3(x - 3)^2 + 1)$ 4. Set $f'(x) = 0$ to find critical points:

• $8(x - 3)^3 = 0 \Rightarrow x = 3$
• Solve $3(x - 3)^2 + 1 = 0$; since this term is always positive, there are no other solutions.

Thus, the only critical point is $x = 3$.

Step 2: Second Derivative for Concavity and Inflection Points

To determine concavity and identify any points of inflection, we calculate $f''(x)$.

1. Differentiate $f'(x) = 8(x - 3)^3(3(x - 3)^2 + 1)$ again.
2. After simplifying, check the sign of $f''(x)$ around $x = 3$.

From the calculations, you’ll find:

• $f(x)$ is concave up on $(-\infty, \infty)$, as $f''(x) > 0$ everywhere.
• There is no interval where $f(x)$ is concave down.
• No inflection points, as the concavity does not change.

Final Answers

1. Point(s) of Local Minimum: $(3, f(3))$
2. Point(s) of Local Maximum: DNE (Does Not Exist)
3. Concave Up Interval: $(-\infty, \infty)$
4. Concave Down Interval: $\varnothing$ (Empty Set)
5. Points of Inflection: DNE (Does Not Exist)

Would you like further details on any of these steps or assistance with related concepts?

Here are five related questions for further practice:

1. How do you determine whether a critical point is a local maximum or minimum?
2. What are the conditions for a function to have an inflection point?
3. How does factoring help in solving higher-degree polynomial equations?
4. How can you use the second derivative test to determine concavity?
5. When would a function have no local extrema or inflection points?

Tip: For polynomial functions, factoring is a powerful method for finding roots and simplifying derivatives for critical point analysis.