Question

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}+3 x+2 )
( f^{prime}(x)=3 x^{2}+3 )
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function is increasing on ( (-infty, 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.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function is decreasing on
(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.

Explanation:

Step1: Analyze the derivative

Given \(f(x)=x^{3}+3x + 2\), its derivative \(f^{\prime}(x)=3x^{2}+3\). Since \(x^{2}\geq0\) for all real \(x\), then \(3x^{2}+3 = 3(x^{2}+1)\geq3>0\) for all \(x\in(-\infty,\infty)\).

Step2: Determine increasing/decreasing intervals

When \(f^{\prime}(x)>0\), the function \(f(x)\) is increasing. As \(f^{\prime}(x)=3x^{2}+3>0\) for all \(x\in(-\infty,\infty)\), the function is increasing on \((-\infty,\infty)\) and never decreasing.

Step3: Check for local extrema

Local extrema occur where \(f^{\prime}(x) = 0\). Solving \(3x^{2}+3=0\) gives \(x^{2}=- 1\), which has no real solutions. So, there are no critical points, and thus no local extrema.

Answer:

A. The function is increasing on \((-\infty,\infty)\)
B. The function is never decreasing
D. The function has no local extrema.