Let \[g(x)=3x^5-30x^4\].

**For what values of \[x\] does the graph of \[g\] have a point of inflection?**

To determine the values of \( x \) where the graph of \( g(x) = 3x^5 - 30x^4 \) has a point of inflection, we need to follow these steps:

### 1. Find the First and Second Derivatives
First, we find the first derivative \( g'(x) \):

\[
g'(x) = \frac{d}{dx}[3x^5 - 30x^4] = 15x^4 - 120x^3
\]

Next, we find the second derivative \( g''(x) \):

\[
g''(x) = \frac{d}{dx}[15x^4 - 120x^3] = 60x^3 - 360x^2
\]

### 2. Set the Second Derivative Equal to Zero
To find potential inflection points, we set the second derivative equal to zero:

\[
60x^3 - 360x^2 = 0
\]

### 3. Factor the Equation
Factor the equation to solve for \( x \):

\[
60x^2(x - 6) = 0
\]

This gives us two solutions:

\[
x^2 = 0 \quad \text{or} \quad x - 6 = 0
\]

So, \( x = 0 \) or \( x = 6 \).

### 4. Confirm Inflection Points by Checking the Sign of \( g''(x) \)
We need to check the sign of \( g''(x) \) on either side of each critical point \( x = 0 \) and \( x = 6 \) to determine if these points are indeed inflection points.

- For \( x = 0 \):
  - Pick a test point less than 0 (e.g., \( x = -1 \)):
    \[
    g''(-1) = 60(-1)^3 - 360(-1)^2 = -60 - 360 = -420 \quad (\text{negative})
    \]
  - Pick a test point greater than 0 but less than 6 (e.g., \( x = 1 \)):
    \[
    g''(1) = 60(1)^3 - 360(1)^2 = 60 - 360 = -300 \quad (\text{negative})
    \]
  Since the sign of \( g''(x) \) does not change around \( x = 0 \), \( x = 0 \) is **not** an inflection point.

- For \( x = 6 \):
  - Pick a test point less than 6 (e.g., \( x = 5 \)):
    \[
    g''(5) = 60(5)^3 - 360(5)^2 = 60(125) - 360(25) = 7500 - 9000 = -1500 \quad (\text{negative})
    \]
  - Pick a test point greater than 6 (e.g., \( x = 7 \)):
    \[
    g''(7) = 60(7)^3 - 360(7)^2 = 60(343) - 360(49) = 20580 - 17640 = 2940 \quad (\text{positive})
    \]
  Since the sign of \( g''(x) \) changes from negative to positive around \( x = 6 \), \( x = 6 \) is an inflection point.

### **Conclusion:**
The graph of \( g(x) = 3x^5 - 30x^4 \) has a point of inflection at \( \mathbf{x = 6} \).

Do you want more details or have any questions? Here are some related questions you might find useful:

1. What is the significance of an inflection point on a graph?
2. How does the second derivative test help in determining concavity?
3. What are the other types of critical points besides inflection points?
4. Can a function have more than one inflection point?
5. How can we use the first derivative to confirm the nature of critical points?

**Tip:** Remember that an inflection point occurs where the second derivative changes sign, indicating a change in the concavity of the function.

Explore how to find points of inflection for the function g(x) = 3x^5 - 30x^4 using calculus. Learn step-by-step methods to determine where the concavity changes, involving the first and second derivatives. Suitable for advanced high school students and calculus enthusiasts.

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