for what values of $x$ is the graph of $y = xe^{-3x}$ concave down? values = (give your answer as an…

for what values of $x$ is the graph of $y = xe^{-3x}$ concave down? values = (give your answer as an interval or a list of intervals, e.g., (-infinity,8 or (1,5),(7,10) )

for what values of $x$ is the graph of $y = xe^{-3x}$ concave down? values = (give your answer as an interval or a list of intervals, e.g., (-infinity,8 or (1,5),(7,10) )

Answer

Explanation:

Step1: Find the first - derivative

Use the product rule $(uv)' = u'v+uv'$, where $u = x$ and $v=e^{-3x}$. $u' = 1$, $v'=- 3e^{-3x}$. Then $y'=e^{-3x}-3xe^{-3x}=e^{-3x}(1 - 3x)$.

Step2: Find the second - derivative

Again use the product rule on $y'=e^{-3x}(1 - 3x)$. Let $u = 1-3x$, $v = e^{-3x}$, $u'=-3$, $v'=-3e^{-3x}$. $y''=-3e^{-3x}-3e^{-3x}(1 - 3x)=e^{-3x}(-3-3 + 9x)=e^{-3x}(9x - 6)$.

Step3: Determine when the function is concave down

A function is concave down when $y''<0$. Since $e^{-3x}>0$ for all real $x$, we solve the inequality $9x - 6<0$. $9x-6<0$ implies $9x<6$, so $x<\frac{2}{3}$.

Answer:

$(-\infty,\frac{2}{3})$