Find the intervals on which $$ f(x) $$ is increasing, the intervals on which $$ f(x) $$ is decreasing, and the local extrema. $$ f(x)=x^{3}-27 x+4 $$ A. The function is increasing on $$ (-\infty,-3),(3, \infty) $$ (Type your answer in interval notation. Type integers or simplified fractions. Use a comma to separate answers as needed.) B. The function is never increasing. Aelect the correct choice below and, if necessary, fill in the answer box to complete your choice. (Type your answer in interval notation. Type integers or simplified fractions Use a comma to separate answers as needed.) B. The function is never decreasing Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Type integers or simplified fractions.) A. The function has a local minimum $$ f(\square)=\square $$, and no local maximum. B. The function has a local maximum $$ f(\square)=\square $$ and a local minimum $$ f(\square)=\square $$ C. The function has a local maximum $$ f(\square)=\square $$, and no local minimum. D. The function has no local extrema. D.

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UpStudy AI · Accepted Answer

1. Compute the derivative of the function:
   $$
   f(x)=x^3-27x+4 \quad\implies\quad f'(x)=3x^2-27=3(x^2-9)=3(x-3)(x+3)
   $$

2. Find the critical points by setting $$ f'(x)=0 $$:
   $$
   3(x-3)(x+3)=0 \quad\implies\quad x=-3 \text{ or } x=3.
   $$

3. Determine the sign of $$ f'(x) $$ on each interval:
   - For $$ x< -3 $$ (for example, $$ x=-4 $$):
     $$
     f'(-4)=3(-4-3)(-4+3)=3(-7)(-1)=21>0,
     $$
     so $$ f(x) $$ is increasing on $$(-\infty,-3)$$.
   - For $$ -33 $$ (for example, $$ x=4 $$):
     $$
     f'(4)=3(4-3)(4+3)=3(1)(7)=21>0,
     $$
     so $$ f(x) $$ is increasing on $$(3,\infty)$$.

4. Identify the local extrema:
   - At $$ x=-3 $$: The derivative changes from positive to negative, indicating a local maximum. Calculate:
     $$
     f(-3)=(-3)^3-27(-3)+4=-27+81+4=58.
     $$
   - At $$ x=3 $$: The derivative changes from negative to positive, indicating a local minimum. Calculate:
     $$
     f(3)=3^3-27(3)+4=27-81+4=-50.
     $$

5. Final Answers:
   - Increasing intervals: $$(-\infty,-3),(3,\infty)$$
   - Decreasing interval: $$(-3,3)$$
   - The function has a local maximum $$ f(-3)=58 $$ and a local minimum $$ f(3)=-50 $$.
The function $$ f(x) = x^3 - 27x + 4 $$ is increasing on $$(-\infty, -3)$$ and $$(3, \infty)$$, decreasing on $$(-3, 3)$$, and has a local maximum at $$ f(-3) = 58 $$ and a local minimum at $$ f(3) = -50 $$.

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