- #1
Mr Davis 97
- 1,462
- 44
The problem statement: Show that if ##r_1## and ##r_2## are the distinct real roots of ##x^2 + px + 8 = 0##, then ##r_1 + r_2 > 4 \sqrt{2}##.
We start by noting that ##r_1 r_2 = 8##. Using this relation, we'll find the minimum value of ##r_1 + r_2##. To minimize ##r_1 + r_2##, we need to minimize each term, thus we need to find the smallest values of both ##r_1## and ##r_2## such that ##r_1 r_2 = 8##. The obvious answer is that ##r_1 + r_2## is minimized when ##r_1, r_2 = 2 \sqrt{2}##, so that ##r_1 + r_2 = 2 \sqrt{2} + 2 \sqrt{2} = 4 \sqrt{2}## is the minimum value of the sum. However, we are under the condition that the roots are distinct, thus any other combination of roots would have to be greater than the minimum, showing that ##r_1 + r_2 > 4 \sqrt{2}##.
Does this correctly show what the problem wants? Is there a flaw in the argument, or a better way to make the argument?
We start by noting that ##r_1 r_2 = 8##. Using this relation, we'll find the minimum value of ##r_1 + r_2##. To minimize ##r_1 + r_2##, we need to minimize each term, thus we need to find the smallest values of both ##r_1## and ##r_2## such that ##r_1 r_2 = 8##. The obvious answer is that ##r_1 + r_2## is minimized when ##r_1, r_2 = 2 \sqrt{2}##, so that ##r_1 + r_2 = 2 \sqrt{2} + 2 \sqrt{2} = 4 \sqrt{2}## is the minimum value of the sum. However, we are under the condition that the roots are distinct, thus any other combination of roots would have to be greater than the minimum, showing that ##r_1 + r_2 > 4 \sqrt{2}##.
Does this correctly show what the problem wants? Is there a flaw in the argument, or a better way to make the argument?