### Instruction: In a certain city the rate of taxation is the following: $x\%$ tax is collected for an income of $x$ thousand dollars. What income, in dollars, will yield the greatest take home pay? (Take-home pay  is the income minus the tax on that income.).### Response: This is an interesting problem that involves finding the maximum value of a function.
I want to find the income $x$ that maximizes the take-home pay $T(x)$.
The take-home pay is the income minus the tax on that income.
The tax on an income of $x$ thousand dollars is $x\%$ of $x$ thousand dollars, or $\frac{x}{100} \times x \times 1000 = 10x^2$ dollars.
So the take-home pay function is $T(x) = x \times 1000 - 10x^2 = 1000x - 10x^2$.
To find the maximum value of $T(x)$, I can use calculus or algebra.
I'll try calculus first.
I need to find the derivative of $T(x)$ and set it equal to zero.
The derivative of $T(x)$ is $T'(x) = 1000 - 20x$.
Setting it equal to zero gives $1000 - 20x = 0$, or $x = 50$.
To check if this is a maximum or a minimum, I can use the second derivative test.
The second derivative of $T(x)$ is $T''(x) = -20$.
This is negative for all values of $x$, which means that $T(x)$ is concave down and has a maximum at $x = 50$.