find the intervals on which f is increasing and the intervals on which it is decreasing. f(x)=x³ + 5x select…

find the intervals on which f is increasing and the intervals on which it is decreasing. f(x)=x³ + 5x select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. o a. the function is increasing on the open interval(s) and decreasing on the open interval(s) (simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.) o b. the function is increasing on the open interval(s) the function is never decreasing. (simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.) o c. the function is decreasing on the open interval(s) the function is never increasing. (simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.) o d. the function is never increasing nor decreasing.

find the intervals on which f is increasing and the intervals on which it is decreasing. f(x)=x³ + 5x select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. o a. the function is increasing on the open interval(s) and decreasing on the open interval(s) (simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.) o b. the function is increasing on the open interval(s) the function is never decreasing. (simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.) o c. the function is decreasing on the open interval(s) the function is never increasing. (simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.) o d. the function is never increasing nor decreasing.

Answer

Explanation:

Step1: Find the derivative

Differentiate $f(x)=x^{3}+5x$ using the power - rule. The derivative $f^\prime(x)=3x^{2}+5$.

Step2: Analyze the sign of the derivative

Since $x^{2}\geq0$ for all real $x$, then $3x^{2}\geq0$. So, $f^\prime(x)=3x^{2}+5\geq5>0$ for all real $x$.

Answer:

B. The function is increasing on the open interval $(-\infty,\infty)$. The function is never decreasing.