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+1) e^{3 x} )\na. the function is increasing on\n(type your answer using 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 using 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.\na. the function has a local minimum ( f(quad)= ) and no local maximum.\nb. the function has a local maximum ( f(quad)= ) and no local minimum.\nc. the function has a local maximum ( f(quad)= ) and a local minimum ( f(quad)= ).\n(type exact answers.)

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+1) e^{3 x} )\na. the function is increasing on\n(type your answer using 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 using 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.\na. the function has a local minimum ( f(quad)= ) and no local maximum.\nb. the function has a local maximum ( f(quad)= ) and no local minimum.\nc. the function has a local maximum ( f(quad)= ) and a local minimum ( f(quad)= ).\n(type exact answers.)

Answer

Explanation:

Step1: Find the derivative of (f(x))

Using the product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = x + 1) and (v=e^{3x}). (u^\prime=1), (v^\prime = 3e^{3x}). So (f^\prime(x)=e^{3x}+3(x + 1)e^{3x}=e^{3x}(1 + 3x+3)=e^{3x}(3x + 4))

Step2: Determine where (f^\prime(x)>0) (increasing) and (f^\prime(x)<0) (decreasing)

Since (e^{3x}>0) for all (x\in R). Set (3x + 4>0), then (x>-\frac{4}{3}). Set (3x + 4<0), then (x<-\frac{4}{3})

Step3: Find local extrema

The critical point is at (x =-\frac{4}{3}). Since (f^\prime(x)) changes sign from negative to positive at (x =-\frac{4}{3}) (using the first - derivative test), (f(x)) has a local minimum at (x =-\frac{4}{3})

Answer:

A. The function is increasing on ((-\frac{4}{3},\infty)) B. The function is decreasing on ((-\infty,-\frac{4}{3})) A. The function has a local minimum (f(-\frac{4}{3})=(-\frac{4}{3}+1)e^{3\times(-\frac{4}{3})}=(-\frac{1}{3})e^{-4}=-\frac{1}{3e^{4}}) and no local maximum.