To find the critical points and the intervals of increase and decrease for the function \( f(x) = 3x^4 - 8x^3 - 6x^2 + 24x + 7 \), we will follow these steps:
### Step 1: Find the derivative of \( f(x) \)
The first step is to differentiate the function \( f(x) \).
\[
f'(x) = \frac{d}{dx}(3x^4 - 8x^3 - 6x^2 + 24x + 7)
\]
Using the power rule, we get:
\[
f'(x) = 12x^3 - 24x^2 - 12x + 24
\]
### Step 2: Set the derivative equal to zero to find critical points
Next, we set the derivative equal to zero to find the critical points:
\[
12x^3 - 24x^2 - 12x + 24 = 0
\]
Dividing the entire equation by 12 simplifies it:
\[
x^3 - 2x^2 - x + 2 = 0
\]
### Step 3: Factor the polynomial
Now we will factor the polynomial \( x^3 - 2x^2 - x + 2 \). We can use the Rational Root Theorem to test possible rational roots. Testing \( x = 2 \):
\[
2^3 - 2(2^2) - 2 + 2 = 8 - 8 - 2 + 2 = 0
\]
Since \( x = 2 \) is a root, we can factor \( x - 2 \) out of the polynomial. Performing synthetic division:
\[
\begin{array}{r|rrrr}
2 & 1 & -2 & -1 & 2 \\
& & 2 & 0 & -2 \\
\hline
& 1 & 0 & -1 & 0 \\
\end{array}
\]
This gives us:
\[
x^3 - 2x^2 - x + 2 = (x - 2)(x^2 - 1)
\]
Factoring \( x^2 - 1 \) further, we have:
\[
x^2 - 1 = (x - 1)(x + 1)
\]
Thus, we can write:
\[
x^3 - 2x^2 - x + 2 = (x - 2)(x - 1)(x + 1) = 0
\]
### Step 4: Find the critical points
Setting each factor to zero gives us the critical points:
\[
x - 2 = 0 \quad \Rightarrow \quad x = 2
\]
\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]
Thus, the critical points are \( x = -1, 1, 2 \).
### Step 5: Determine intervals of increase and decrease
To find the intervals of increase and decrease, we will test the sign of \( f'(x) \) in the intervals defined by the critical points: \( (-\infty, -1) \), \( (-1, 1) \), \( (1, 2) \), and \( (2, \infty) \).
1. **Interval \( (-\infty, -1) \)**: Choose \( x = -2 \)
\[
f'(-2) = 12(-2)^3 - 24(-2)^2 - 12(-2) + 24 = -96 - 96 + 24 + 24 = -144 \quad (\text{negative})
\]
2. **Interval \( (-1, 1) \)**: Choose \( x = 0 \)
\[
f'(0) = 12(0)^3 - 24(0)^2 - 12(0) + 24 = 24 \quad (\text{positive})
\]
3. **Interval \( (1, 2) \)**: Choose \( x = 1.5 \)
\[
f'(1.5) = 12(1.5)^3 - 24(1.5)^2 - 12(1.5) + 24 = 12(3.375) - 24(2.25) - 18 + 24 = 40.5 - 54 - 18 + 24 = -7.5 \quad (\text{negative})
\]
4. **Interval \( (2, \infty) \)**: Choose \( x = 3 \)
\[
f'(3) = 12(3)^3 - 24(3)^2 - 12(3) + 24 = 12(27) - 24(9) - 36 + 24 = 324 - 216 - 36 + 24 = 96 \quad (\text{positive})
\]
### Step 6: Summarize the results
- **Increasing** on the intervals: \( (-1, 1) \) and \( (2, \infty) \)
- **Decreasing** on the intervals: \( (-\infty, -1) \) and \( (1, 2) \)
### Conclusion
The critical points are \( x = -1, 1, 2 \). The function is increasing on \( (-1, 1) \) and \( (2, \infty) \) and decreasing on \( (-\infty, -1) \) and \( (1, 2) \).
