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

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.