find the intervals of concavity and inflection points. list all answers in each category separated by…

find the intervals of concavity and inflection points. list all answers in each category separated by commas, and enter none if none exist.\n\n$f(x)=x^{4}+4x^{3}+6x^{2}+4x$\n\nconcave up on:\n\nconcave down on:\n\ninflection points $(x,y):$

find the intervals of concavity and inflection points. list all answers in each category separated by commas, and enter none if none exist.\n\n$f(x)=x^{4}+4x^{3}+6x^{2}+4x$\n\nconcave up on:\n\nconcave down on:\n\ninflection points $(x,y):$

Answer

Explanation:

Step1: Find the first - derivative

Using the power rule $(x^n)'=nx^{n - 1}$, we have $f'(x)=4x^{3}+12x^{2}+12x + 4$.

Step2: Find the second - derivative

Differentiate $f'(x)$ again. $f''(x)=12x^{2}+24x + 12$.

Step3: Factor the second - derivative

$f''(x)=12(x^{2}+2x + 1)=12(x + 1)^{2}$.

Step4: Find the intervals of concavity

Set $f''(x)=0$. Since $12(x + 1)^{2}=0$, we get $x=-1$. Test intervals: Choose a test point in the interval $(-\infty,-1)$, say $x=-2$. Then $f''(-2)=12(-2 + 1)^{2}=12>0$. Choose a test point in the interval $(-1,\infty)$, say $x = 0$. Then $f''(0)=12(0 + 1)^{2}=12>0$. Since $f''(x)\geq0$ for all $x\in(-\infty,\infty)$ (equality holds only at $x=-1$), the function is concave up on $(-\infty,\infty)$ and concave down on NONE.

Step5: Find the inflection points

Since the concavity does not change ( $f''(x)$ does not change sign), the inflection points are NONE.

Answer:

Concave up on: $(-\infty,\infty)$ Concave down on: NONE Inflection points $(x,y)$: NONE