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 )

find ( f^{prime}(x) ).

( 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.

Explanation:

Step1: Analyze the derivative

Given \(f'(x)=3x^{2}+3\). Since \(x^{2}\geq0\) for all real \(x\), then \(3x^{2}+3 = 3(x^{2}+1)\). And \(x^{2}+1>0\) for all real \(x\) (because \(x^{2}\geq0\) implies \(x^{2}+1\geq1\)). So \(f'(x)>0\) for all \(x\in(-\infty,\infty)\).

Step2: Determine increasing and decreasing intervals

When \(f'(x)>0\) on an interval, the function \(f(x)\) is increasing on that interval. When \(f'(x)<0\) on an interval, the function \(f(x)\) is decreasing on that interval. Since \(f'(x)=3x^{2}+3>0\) for all \(x\in(-\infty,\infty)\), the function \(f(x)\) has no intervals where \(f'(x)<0\).

Answer:

For the increasing - decreasing part:

• The function is increasing on \((-\infty,\infty)\) (so option A is correct for the increasing part).
• The function is never decreasing (so option B is correct for the decreasing part).

Since \(f'(x)\) never changes sign (it is always positive), there are no local extrema (because local extrema occur where \(f'(x)\) changes sign from positive to negative or vice - versa).