Identify the open intervals on which the function is increasing or decreasing.
$f(x)=3x^4+8x^3-18x^2+7$

• A. Increasing on $\left(-\infty,-3\right)\&\left(0,1\right)$, Decreasing on $\left(-3,0\right)\&\left(1,\infty\right)$
• B. Increasing on $\left(-3,0\right)\&\left(1,\infty\right)$, Decreasing on $\left(-\infty,-3\right)\&\left(0,1\right)$
• C. Increasing on $\left(-\infty,0\right)$, Decreasing on $\left(0,\infty\right)$
• D. Increasing on $\left(0,\infty\right)$, Decreasing on $\left(-\infty,0\right)$

Answer: (B)

Verified step by step guidance

1. To determine where the function is increasing or decreasing, we first need to find its derivative. The function given is \( f(x) = 3x^4 + 8x^3 - 18x^2 + 7 \).
2. Calculate the derivative \( f'(x) \) using the power rule. The power rule states that the derivative of \( ax^n \) is \( n \cdot ax^{n-1} \). Apply this to each term of the function.
3. The derivative is \( f'(x) = 12x^3 + 24x^2 - 36x \).
4. Find the critical points by setting \( f'(x) = 0 \) and solving for \( x \). This involves factoring the derivative or using the quadratic formula if necessary.
5. Once the critical points are found, use a sign chart or test intervals around these points to determine where \( f'(x) \) is positive (indicating the function is increasing) or negative (indicating the function is decreasing). This will help identify the open intervals where the function is increasing or decreasing.