Science:Math Exam Resources/Courses/MATH110/April 2016/Question 02 (b)

MATH110 April 2016

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Question 02 (b)
Find the coordinates of the inflection point of the function $g(x)=xe^{x}$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
We may use the second derivative to find the inflection points: If $x$ is the inflection point, then $f''(x)=0$ and $f''(x+\epsilon ){\text{and}}f''(x-\epsilon )$ have opposite signs.

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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. $g(x)=xe^{x}$ $g'(x)=e^{x}+xe^{x}$ $g''(x)=e^{x}+e^{x}+xe^{x}=2e^{x}+xe^{x}$ set $g''(x)=0$, we have $2e^{x}+xe^{x}=e^{x}(2+x)=0\rightarrow x=-2$, and in the neighborhood of -2, $g''(x)$ have opposite signs. Therefore, when $x=-2\rightarrow g(-2)=-2e^{-2}$ answer: $\color {blue}(-2,-2e^{-2})$