determine where the function is concave upward and where it is concave downward. (enter your answers using…

Explanation:

Step1: Find the first - derivative

Using the power rule $(x^n)'=nx^{n - 1}$, for $f(x)=3x^{4}-30x^{3}+x - 7$, we have $f'(x)=12x^{3}-90x^{2}+1$.

Step2: Find the second - derivative

Differentiating $f'(x)$ again using the power rule, $f''(x)=36x^{2}-180x=36x(x - 5)$.

Step3: Find the inflection points

Set $f''(x)=0$, so $36x(x - 5)=0$. Solving this equation gives $x = 0$ and $x = 5$.

Step4: Test the intervals

Choose test points in the intervals $(-\infty,0)$, $(0,5)$ and $(5,\infty)$. For the interval $(-\infty,0)$, let $x=-1$. Then $f''(-1)=36\times(-1)\times(-1 - 5)=216>0$. For the interval $(0,5)$, let $x = 1$. Then $f''(1)=36\times1\times(1 - 5)=-144<0$. For the interval $(5,\infty)$, let $x = 6$. Then $f''(6)=36\times6\times(6 - 5)=216>0$.

Answer:

concave upward: $(-\infty,0)\cup(5,\infty)$ concave downward: $(0,5)$