Understanding the Distinction Between Maximum and Minimum Values

Chapter 1: Weekly Challenge Recap

This week's challenge appears to be a bit more straightforward compared to previous ones. Regardless, let's give a round of applause to this week's top performers! 1st Place: Alfredo Esposito 7 Points 🎉 2nd Place: Neil Faver 5 Points 🎉 3rd Place: Wotsit 3 Points 🎉 Other Participants: Aaron Tovish, Jonathan Lynn Harvey 1 Point each 🎉

Here’s the solution! We can start by rearranging the two equations to consolidate all terms involving 'a' and 'b' on one side.

Rearranged equations for max-min calculations.

The key to solving this problem is to embrace the idea of treating 'x' as a variable.

The concept of using 'x' as a variable in equations.

This inequality holds true since we are summing two squared terms, and each squared term must be zero or greater. Upon expanding the terms, we derive the following quadratic inequality.

Note that the expressions (a + b) and (a² + b²) can be expressed in terms of 'c'.

Now we have a quadratic inequality that is greater than or equal to zero, with coefficients expressed in terms of 'c'. Remember from high school algebra that a quadratic equation has two distinct real roots if the discriminant is greater than zero, one repeated root if it equals zero, and no real roots if it is less than zero. Our quadratic equation must be greater than or equal to zero, indicating that it either has no real roots or it touches the x-axis at one point. Hence, we can conclude that its discriminant must be less than or equal to zero.