Science:Math Exam Resources/Courses/MATH110/April 2014/Question 03 (f)

MATH110 April 2014

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Question 03 (f)
Let $f(x)={\dfrac {x^{2}}{(x-4)^{2}}}$. Find the intervals where $f$ is concave up and the intervals where it 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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
The second derivative gives us information about the concavity of the function.

Hint 2
Partition the function according to the points where $f''(x)=0$ or is undefined.

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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. The second derivative is $f''(x)={\frac {16(x+2)}{(x-4)^{4}}}.$ The point of interest in this case is $x=-2$. So we need to partition the domain and test values within the intervals. Interval $f''(x)$ CU/CD $(-\infty ,-2)$ $f''(-10)<0$ CD $(-2,4)$ $f''(0)>0$ CU $(4,\infty )$ $f''(10)>0$ CU The function is concave up on $(-2,4)$ and $(4,\infty )$ and concave down on $(-\infty ,-2)$.

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