To punish his class for indiscipline, Professor Zerus decided to deduct from each student's monthly grade a percentage equal to the test score, that is: whoever scored 60 will have a 60% deduction from their grade, whoever scored 20 will have a 20% deduction, and so on. The maximum monthly grade is 100.

(a) Who will end up with the highest grade?

(b) And the lowest?

(c) Students who scored well complained that they will end up with the same grade as those who scored poorly. Are they right?
Whoever had $x$ as their monthly grade will receive a discount of $x \%$ on that grade, meaning they will lose

$$
x \% \text { of } x=\frac{x}{100} \times x=\frac{x^{2}}{100}
$$

Thus, after the punishment, the grade becomes $x-\frac{x^{2}}{100}$, where $x$ was the initial grade.

Consider the function "grade after the punishment" given by $f(x)=x-\frac{x^{2}}{100}$. Since the maximum and minimum grades are 0 and 100, we will consider this function in the domain $[0,100]$, i.e., for $0 \leq x \leq 100$. The graph of $f$ is a parabola with concavity downwards, and its maximum value occurs at the vertex: $x=-\frac{b}{2 a}=\frac{-\frac{1}{\frac{-2}{100}}}{}=50$. Therefore, the highest grade after the punishment is for students who had 50 before the punishment. This grade is

$$
f(50)=50-\frac{50^{2}}{100}=25
$$

The minimum value of this function is 0 and occurs at $x=0$ and $x=100$. Thus, the lowest grade occurs for students who had 0 or 100(!!!!!) before the punishment. Indeed, $f(0)=f(100)=0$.

![](https://cdn.mathpix.com/cropped/2024_05_01_a444074f1f8ae50444d5g-30.jpg?height=380&width=807&top_left_y=1390&top_left_x=583)