find the intervals on which f is increasing and the intervals on which it is decreasing. f(x)=x^4/4 + x^3 +…

find the intervals on which f is increasing and the intervals on which it is decreasing. f(x)=x^4/4 + x^3 + x^2 select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. a. the function is increasing on the open interval(s) . the function is never decreasing. (simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.) b. the function is decreasing on the open interval(s) and increasing on the open interval(s) (simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.) c. the function is decreasing on the open interval(s) . the function is never increasing. (simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.) d. the function is never increasing or decreasing.

find the intervals on which f is increasing and the intervals on which it is decreasing. f(x)=x^4/4 + x^3 + x^2 select the correct choice and, if necessary, fill in the answer box(es) to complete your choice. a. the function is increasing on the open interval(s) . the function is never decreasing. (simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.) b. the function is decreasing on the open interval(s) and increasing on the open interval(s) (simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.) c. the function is decreasing on the open interval(s) . the function is never increasing. (simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.) d. the function is never increasing or decreasing.

Answer

Explanation:

Step1: Find the derivative

Differentiate $f(x)=\frac{x^{4}}{4}+x^{3}+x^{2}$ using power - rule. The derivative $f'(x)=x^{3}+3x^{2}+2x=x(x + 1)(x + 2)$.

Step2: Find the critical points

Set $f'(x)=0$. So $x(x + 1)(x + 2)=0$. The critical points are $x=-2,x=-1,x = 0$.

Step3: Test the intervals

Consider the intervals $(-\infty,-2),(-2,-1),(-1,0),(0,\infty)$. For $x\in(-\infty,-2)$, let $x=-3$. Then $f'(-3)=(-3)(-3 + 1)(-3 + 2)=(-3)(-2)(-1)=-6<0$, so $f(x)$ is decreasing on $(-\infty,-2)$. For $x\in(-2,-1)$, let $x =-\frac{3}{2}$. Then $f'(-\frac{3}{2})=(-\frac{3}{2})(-\frac{3}{2}+1)(-\frac{3}{2}+2)=(-\frac{3}{2})(-\frac{1}{2})(\frac{1}{2})=\frac{3}{8}>0$, so $f(x)$ is increasing on $(-2,-1)$. For $x\in(-1,0)$, let $x=-\frac{1}{2}$. Then $f'(-\frac{1}{2})=(-\frac{1}{2})(-\frac{1}{2}+1)(-\frac{1}{2}+2)=(-\frac{1}{2})(\frac{1}{2})(\frac{3}{2})=-\frac{3}{8}<0$, so $f(x)$ is decreasing on $(-1,0)$. For $x\in(0,\infty)$, let $x = 1$. Then $f'(1)=(1)(1 + 1)(1 + 2)=(1)(2)(3)=6>0$, so $f(x)$ is increasing on $(0,\infty)$.

Answer:

B. The function is decreasing on the open intervals $(-\infty,-2),(-1,0)$ and increasing on the open intervals $(-2,-1),(0,\infty)$