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. Do they have a point?
Those who had a monthly grade of $x$ 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, a grade initially of $x$, after the penalty, becomes $x-\frac{x^{2}}{100}$. Let's consider this "grade after the penalty" function, given by

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

Since the maximum and minimum grades are 0 and 100, we can consider this function only in the domain $[0,100]$, that is, for $0 \leq x \leq 100$. The graph of $f$ is a parabola with concavity downwards. The minimum value of this function is 0, which occurs at $x=0$ and $x=100$, and the maximum value occurs at the vertex, that is, at the point $x=50$, which is the arithmetic mean between the two roots 0 and 100 of $f$.

(a) The highest grade after the penalty is for students who, before the penalty, scored 50. This grade is

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

(b) The lowest grade is 0 and occurs for students who scored 0 or, surprisingly, 100 before the penalty. Indeed, $f(0)=f(100)=0$.

(c) For each grade greater than 50, there is another grade, less than 50, that ends up being the same after the penalty. Indeed, by the symmetry of the parabola, $f(50-n)=f(50+n)$, for each $0 \leq n \leq 50$. For example, those who scored 30 end up with the same grade of 21 as those who scored 70, since $f(30)=21=f(70)$. Thus, the complaints of students who scored well are justified.