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 + 3x + 8 find f(x). f(x)=x^3 + 3x + 8 f(x)= 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. 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.)

Explanation:

Step1: Differentiate the function

Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, for $f(x)=x^{3}+3x + 8$, we have $f'(x)=3x^{2}+3$.

Step2: Analyze the sign of the derivative

Factor out 3 from $f'(x)$: $f'(x)=3(x^{2}+1)$. Since $x^{2}\geq0$ for all real $x$, then $x^{2}+1>0$ for all real $x$, and $f'(x)=3(x^{2}+1)>0$ for all real $x$.

Answer:

$f'(x)=3x^{2}+3$
A. The function is increasing on $(-\infty,\infty)$
B. The function is never decreasing. There are no local extrema since the function is always increasing.