Science:Math Exam Resources/Courses/MATH100 C/December 2024/Question 12 (b)

{{#incat:MER QGQ flag|{{#incat:MER QGH flag|{{#incat:MER QGS flag|}}}}}}

MATH100 C December 2024

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q3 (a)  •  Q3 (b)  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q5  •  Q6  •  Q7  •  Q8  •  Q9 (a)  •  Q9 (b)  •  Q10 (a)  •  Q10 (b)  •  Q10 (c)  •  Q11 (a)  •  Q11 (b)  •  Q11 (c)  •  Q12 (a)  •  Q12 (b)  •  Q12 (c)  •  Q12 (d)  •  Q13  •  Q14  •  

   Other MATH100 C Exams

  •  December 2023  •  December 2024  •  

Question 12 (b)

Consider an income tax system in which the tax $T$ paid on an income $x$ is given by the formula $$T(x)=\frac{x^3}{1+2x^2}$$ where both $x$ and $T$ are measured in units of $\mathrm{\$}10,000.$ It is given that $$T^{\prime }(x)=\frac{x^2(2x^2+3)}{(1+2x^2{)}^2},\text{and},T^{″}(x)=\frac{2x(3-2x^2)}{(1+2x^2{)}^3}.$$ Show that income after taxes (that is, $x-T(x)$) is an increasing function of income.

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
Differentiate $x-T(x)$ and use the given formula for $T^{\prime }(x)$. Show the derivative is always positive.

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. 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

Consider income after taxes: $$f(x)=x-T(x).$$ Differentiate: $$f^{\prime }(x)=1-T^{\prime }(x).$$ Using the given expression for $T^{\prime }(x)$: $$f^{\prime }(x)=1-\frac{x^2(2x^2+3)}{(1+2x^2{)}^2}.$$ Write over a common denominator: $$f^{\prime }(x)=\frac{(1+2x^2{)}^2-x^2(2x^2+3)}{(1+2x^2{)}^2}.$$ Expanding and simplifying the expression above, we get: $$f^{\prime }(x)=\frac{1+x^2+2x^4}{(1+2x^2{)}^2}.$$ The numerator $1+x^2+2x^4$ is always positive, and the denominator $(1+2x^2{)}^2$ is always positive. Hence, $$f^{\prime }(x)>0\text{for all }x.$$ Therefore, $x-T(x)$ is an increasing function of income.

{{#incat:MER CT flag||

Click here for similar questions

MER CT flag, MER QGH flag, MER QGQ flag, MER QGS flag, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag

}}