Science:Math Exam Resources/Courses/MATH104/December 2015/Question 10 (c)

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MATH104 December 2015

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Question 10 (c)
Let $f(x)=x^{2}+x-{\frac {1}{x}}-1$. Its derivative satisfies $f'(x)=2x+1+{\frac {1}{x^{2}}}={\frac {2x^{3}+x^{2}+1}{x^{2}}}={\frac {(x+1)(2x^{2}-x+1)}{x^{2}}}$. (c) Find the intervals on which $f$ is concave up. Hint: Think about what version of $f'(x)$ you would like to differentiate!

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
Recall that a (twice-differentiable) function $f(x)$ is concave up (or convex) on an interval if $f''(x)\geq 0$ on that interval.

Hint 2
For simplicity, avoid using the product/quotient rules if possible.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

To determine the intervals of convexity/concavity of $f$, we consider the sign of $f''(x)={\frac {d}{dx}}\left(2x+1+{\frac {1}{x^{2}}}\right)=2-{\frac {2}{x^{3}}}=2\left(1-{\frac {1}{x^{3}}}\right)\,.$ Now $f''$ may change sign across its discontinuity at $x=0$ and/or at its root $x=1.$ Checking this, we find that $f''(x){\color {OrangeRed}>0}$ when $x<0$ $f''(x){\color {OliveGreen}<0}$ when $0<x<1$ $f''(x){\color {OrangeRed}>0}$ when $x>1$ Or, in tabular form, Interval: $(-\infty ,0)$ $(0,1)$ $(1,\infty )$ Sign of $2$ ${\color {OrangeRed}+}$ ${\color {OrangeRed}+}$ ${\color {OrangeRed}+}$ Sign of $1-1/x^{3}$ ${\color {OrangeRed}+}$ ${\color {OliveGreen}-}$ ${\color {OrangeRed}+}$ Sign of $f''(x)$ ${\color {OrangeRed}+}\cdot {\color {OrangeRed}+}={\color {OrangeRed}+}$ ${\color {OrangeRed}+}\cdot {\color {OliveGreen}-}={\color {OliveGreen}-}$ ${\color {OrangeRed}+}\cdot {\color {OrangeRed}+}={\color {OrangeRed}+}$ Since a (twice-differentiable) function $f(x)$ is concave up (convex) on an interval if $f''(x)\geq 0$ on that interval, we have that $f(x)$ is concave up on ${\color {blue}(-\infty ,0)\cup [1,\infty )}.$ Note that we have excluded $0$ from the interval as $f$ is not even defined at $x=0.$

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