Science:Math Exam Resources/Courses/MATH100 C/December 2024/Question 12 (b)/Solution 1

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.