find the intervals on which ( f(x) ) is increasing, the intervals on which ( f(x) ) is decreasing, and the…

find the intervals on which ( f(x) ) is increasing, the intervals on which ( f(x) ) is decreasing, and the local extrema.\n( f(x)=x^{3}+5 x+6 )\nfind ( f^{prime}(x) ).\n( f(x)=x^{3}+5 x+6 )\n( f^{prime}(x)=square )\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function is increasing on \n(type your answer in interval notation. type integers or simplified fractions. use a comma to separate answers as needed.)\nb. the function is never increasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function is decreasing on \n(type your answer in interval notation. type integers or simplified fractions. use a comma to separate answers as needed.)\nb. the function is never decreasing.\nselect the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n(type integers or simplified fractions.)

find the intervals on which ( f(x) ) is increasing, the intervals on which ( f(x) ) is decreasing, and the local extrema.\n( f(x)=x^{3}+5 x+6 )\nfind ( f^{prime}(x) ).\n( f(x)=x^{3}+5 x+6 )\n( f^{prime}(x)=square )\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function is increasing on \n(type your answer in interval notation. type integers or simplified fractions. use a comma to separate answers as needed.)\nb. the function is never increasing.\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the function is decreasing on \n(type your answer in interval notation. type integers or simplified fractions. use a comma to separate answers as needed.)\nb. the function is never decreasing.\nselect the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n(type integers or simplified fractions.)

Answer

Explanation:

Step1: Differentiate the function

Using the power rule ((x^n)^\prime = nx^{n - 1}), for (f(x)=x^{3}+5x + 6), we have (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 (3x^{2}+5\geq5>0) for all (x\in R).

Answer:

(f^\prime(x)=3x^{2}+5); The function is never decreasing (Option B); The function is increasing on ((-\infty,\infty)) (Option A)