Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 11 (d)

MATH 180 December 2017

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Question 11 (d)
The function $$P(x)={\frac {1}{4}}xe^{-x/2},\quad x\geq 0,$$ is an example of a Gamma distribution, a function that may be used to describe certain probabilities. (d) The second derivative of ${\textstyle P(x)}$ is $$P''(x)={\frac {1}{16}}e^{-x/2}(x-4).$$ Identify all the intervals where ${\textstyle P(x)}$ is concave up, and all the intervals where ${\textstyle P(x)}$ is concave down.

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
Recall that a function is concave up if its second derivative is positive, and is concave down if its second derivative is negative.

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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. We proceed similarly as in part (c). By the Hint, we consider the sign of ${\textstyle P''(x)}$, which is the same as the sign of $$x-4.$$ Clearly, this is positive when ${\textstyle x>4}$ and is negative when ${\textstyle x<4}$. Therefore, ${\textstyle P(x)}$ is concave up on the interval where ${\textstyle x>4}$, that is on ${\textstyle (4,\infty )}$. Similarly, ${\textstyle P(x)}$ is concave down on the interval where ${\textstyle x<4}$, or ${\textstyle [0,4)}$. Answer: ${\textstyle P(x)}$ is ${\textstyle {\color {blue}{\mbox{concave up on }}(4,\infty )}}$ and ${\textstyle {\color {blue}{\mbox{concave down on }}[0,4)}}$.