Final answer:
The function y = x² – 3x² - 6x + 8 simplifies to y = -2x² - 6x + 8. Its relative maximum is found at the vertex of the parabola which is (1.5, -4.5). There is no relative minimum since the parabola opens downwards.
Step-by-step explanation:
To find the relative maximum and minimum of the function y = x² – 3x² - 6x + 8, we first need to simplify the expression. Notice that x² and –3x² are like terms, so the equation simplifies to y = -2x² - 6x + 8. This is a quadratic function, and its graph is a parabola opening downwards (since the coefficient of x² is negative).
To find the vertex of the parabola, which will give us the relative maximum because the parabola opens downwards, we use the vertex formula x = -b/(2a). Here, a = -2 and b = -6, so the x-coordinate of the vertex is x = -(-6)/(2 × -2) = 1.5. To find the y-coordinate of the vertex, we substitute x back into the equation: y = -2(1.5)² - 6 × 1.5 + 8 = -4.5. Therefore, the vertex of the parabola is at (1.5, -4.5), and this point is also the relative maximum.
As for the relative minimum, since this parabola opens downwards and thus has no minimum point within its domain, we can conclude there is no relative minimum in this function for all real numbers.