Science:Math Exam Resources/Courses/MATH102/December 2016/Question A 01

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MATH102 December 2016

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Question A 01
Suppose that you have conducted an experiment resulting in the data plotted below. The following functions $f(x)$ depend on a parameter $b>0$ . Which function would you use to fit the data points above? (i) $f(x)={\frac {8e^{x}}{b+e^{x}}}$ (ii) $f(x)={\frac {8x}{b+x}}$ (iii) $f(x)={\frac {8x^{3}}{b^{3}+x^{3}}}$ (iv) $f(x)={\frac {bx^{3}}{8^{3}+x^{3}}}$

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
Observe that $f(0)=0$ and $\lim _{x\to \infty }f(x)=8$ . Also, consider the concavity of the function.

Checking a solution serves two purposes: helping you if, after having used the hint, 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.
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Solution
From the graph, we can easily see that $f(0)=0$ and $\lim _{x\to \infty }f(x)=8$ . Furthermore, the function is concave up on $(0,2)$ and concave down on $(2,\infty )$ . Using this information, (i) can't be the answer because of $f(0)={\frac {8e^{0}}{b+e^{0}}}={\frac {8}{b+1}}\neq 0$ . On the other hand, (iv) is also not the answer because $\lim _{x\to \infty }f(x)=\lim _{x\to \infty }{\frac {bx^{3}}{8^{3}+x^{3}}}=b$ . ( $b\neq 8$ in general) Finally, we exclude (ii), which follows from $f''(x)=\left({\frac {8x}{b+x}}\right)''=\left(8-{\frac {8b}{b+x}}\right)''=\left({\frac {8b}{(b+x)^{2}}}\right)'=-2\cdot {\frac {8b}{(b+x)^{3}}}<0$ for any $x\geq 0$ . Therefore, the answer is $\color {blue}(iii)$