Question

При каких значениях m вершины парабол y=-x2+4mx-m и y=x2+2mx-2 расположены по одну сторону от оси x?

Answer 1

Вершина параболы имеет координату по x = -b/(2a), то есть
x1 = -4m/(-2) = 2m, x2 = -2m/2 = -m
А координаты по y
y1 = -(2m)^2 + 4m*2m - m = -4m^2 + 8m^2 - m = 4m^2 - m
y2 = (-m)^2 + 2m(-m) - 2 = m^2 - 2m^2 - 2 = -m^2 - 2
Если они по одну сторону от оси х, то y1 и y2 имеют одинаковые знаки.

1) Обе вершины расположены ниже оси х.
{ 4m^2 - m < 0
{ -m^2 - 2 < 0
Получаем
{ m(4m - 1) < 0
{ m^2 + 2 > 0 - это верно при любом m
0 < m < 1/4

2) Обе вершины расположены выше оси х
{ 4m^2 - m > 0
{ -m^2 - 2 > 0 - это не верно ни при каком m
Решений нет

Answer 2

Understanding the Position of Parabolas

When considering the position of the parabolas y = -x^2 + 4mx - m and y = x^2 + 2mx - 2 in relation to the x-axis, we need to determine the values of m for which the parabolas are located on the same side of the x-axis.

Position of Parabolas Based on the Value of m

The position of the parabolas in relation to the x-axis is determined by the discriminant of the quadratic equation. For the parabola y = -x^2 + 4mx - m, the discriminant is 16m^2 - 4(1)(-m) = 16m^2 + 4m. For the parabola y = x^2 + 2mx - 2, the discriminant is 4m^2 + 8.

To determine when the parabolas are on the same side of the x-axis, we need to find the values of m for which the discriminants have the same sign.

Finding the Values of m

By setting the discriminants equal to each other and solving for m, we can find the values of m for which the parabolas are on the same side of the x-axis.

Setting the discriminants equal to each other: 16m^2 + 4m = 4m^2 + 8

Solving for m: 16m^2 + 4m - 4m^2 - 8 = 0 12m^2 + 4m - 8 = 0

Using the quadratic formula: m = (-b ± √(b^2 - 4ac)) / (2a) where a = 12, b = 4, and c = -8

m = (-4 ± √(4^2 - 4(12)(-8))) / (2*12) m = (-4 ± √(16 + 384)) / 24 m = (-4 ± √400) / 24 m = (-4 ± 20) / 24

This gives us two potential values for m. By analyzing the sign of the discriminants for these values of m, we can determine when the parabolas are on the same side of the x-axis.

Conclusion

Based on the analysis, the values of m for which the parabolas y = -x^2 + 4mx - m and y = x^2 + 2mx - 2 are located on the same side of the x-axis are when m = -1 and m = 1/3.