Question

find the intervals on which f(x) is increasing, the intervals on which f(x) is decreasing and the local extrema.

f(x)=x³ + 3x + 2

find f(x).

f(x) = x³ + 3x + 2

f(x) = 3x² + 3

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

○ a. the function is increasing 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 increasing.

Explanation:

Step1: Analyze the derivative

Given \( f^{\prime}(x)=3x^{2}+3 \). Since \( x^{2}\geq0 \) for all real \( x \), then \( 3x^{2}\geq0 \). Adding 3, we get \( 3x^{2}+3\geq3>0 \) for all \( x\in R \).

Step2: Determine increasing/decreasing intervals

If \( f^{\prime}(x)>0 \) on an interval \( I \), then \( f(x) \) is increasing on \( I \). Since \( f^{\prime}(x) = 3x^{2}+3>0 \) for all \( x\in(-\infty,\infty) \), the function \( f(x) \) has no intervals where it is decreasing. Also, since the derivative never changes sign (it is always positive), there are no local extrema.

Answer:

A. The function is increasing on \( (-\infty,\infty) \)