Science:Math Exam Resources/Courses/MATH100/December 2015/Question 10 (h)

MATH100 December 2015

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Question 10 (h)
Let $f(x)=(x+2)(x+1)e^{-x}$. (h) The second derivative is $f''(x)=(x-2)(x+1)e^{-x}$. Find the x-coordinates of any points of inflection. You must explain your answer.

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Hint
Inflection points of a function $f(x)$ are the points at which $f''(x)=0$ and $f''(x)$ changes sign in their neighborhood.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. First, we should find the points where $f''(x)=0$. $f''(x)=(x-2)(x+1)e^{-x}=0\Rightarrow x-2=0\quad {\text{or}}\quad x+1=0\quad {\text{or}}\quad e^{-x}=0$ Again the last equation never happens, so $x=2,x=-1$. The next step is to check the sign of $f''(x)$: $e^{-x}$ is always positive, so it doesn't change anything in the sign. We check the sign of $f''(x)$ for $x$'s such that $x<-1,\ -1<x<2,\ x>2$, for example $-2,0,3$ which results in the following sign chart: $\displaystyle (-\infty ,-1)$ $\displaystyle (-1,2)$ $\displaystyle (2,\infty )$ $x-2$ $\displaystyle -$ $\displaystyle -$ $\displaystyle +$ $x+1$ $\displaystyle -$ $\displaystyle +$ $\displaystyle +$ $e^{-x}$ $\displaystyle +$ $\displaystyle +$ $\displaystyle +$ $f''(x)=e^{-x}(x-2)(x+1)$ $\displaystyle +$ $\displaystyle -$ $\displaystyle +$ $f(x)$ concave up concave down concave up Intervals that $f(x)$ is concave up: ${\color {blue}(-\infty ,-1)\ (2,\infty )}$. Intervals that $f(x)$ is concave down: ${\color {blue}(-1,2)}$. Inflection points: ${\color {blue}x=-1,x=2}$.