determine the intervals on which the following function is concave up or concave down. identify any…

Explanation:

Step1: Find the first - derivative

Use the product rule $(uv)' = u'v+uv'$, where $u=-e^{x}$ and $v = x - 10$. $u'=-e^{x}$ and $v' = 1$. $f'(x)=-e^{x}(x - 10)-e^{x}=-e^{x}(x - 10 + 1)=-e^{x}(x - 9)$.

Step2: Find the second - derivative

Again use the product rule. Let $u=-e^{x}$ and $v=x - 9$. $u'=-e^{x}$ and $v' = 1$. $f''(x)=-e^{x}(x - 9)-e^{x}=-e^{x}(x - 9+1)=-e^{x}(x - 8)$.

Step3: Find inflection points

Set $f''(x)=0$. Since $e^{x}>0$ for all $x\in R$, then $x - 8 = 0$, so $x = 8$.

Step4: Determine concavity

Test intervals: Choose a test - point in the interval $(-\infty,8)$, say $x = 0$. Then $f''(0)=-e^{0}(0 - 8)=8>0$, so the function is concave up on $(-\infty,8)$. Choose a test - point in the interval $(8,\infty)$, say $x = 9$. Then $f''(9)=-e^{9}(9 - 8)=-e^{9}<0$, so the function is concave down on $(8,\infty)$.

Answer:

A. The function is concave up on $(-\infty,8)$ and concave down on $(8,\infty)$.