x^3-6x^2>0
anser in interval notation
To solve the inequality x^3 - 6x^2 > 0, we can start by factoring out common terms from both sides:
x^2(x - 6) > 0
Next, we can identify the critical points where the expression equals zero. In this case, we have x^2 = 0 and x - 6 = 0. Solving these equations, we find that x = 0 and x = 6.
Now we have three intervals to consider: (-∞, 0), (0, 6), and (6, ∞).
For the interval (-∞, 0), we can choose a test point, say x = -1. Plugging this value into the expression x^2(x - 6), we get (-1)^2(-1 - 6) = (-1)(-7) = 7. Since 7 > 0, the inequality x^3 - 6x^2 > 0 is true in this interval.
For the interval (0, 6), we can choose another test point, say x = 1. Plugging this value into the expression x^2(x - 6), we get (1)^2(1 - 6) = (1)(-5) = -5. Since -5 < 0, the inequality x^3 - 6x^2 > 0 is false in this interval.
For the interval (6, ∞), we can choose a third test point, say x = 7. Plugging this value into the expression x^2(x - 6), we get (7)^2(7 - 6) = (49)(1) = 49. Since 49 > 0, the inequality x^3 - 6x^2 > 0 is true in this interval.
Therefore, the solution to the inequality x^3 - 6x^2 > 0, in interval notation, is (-∞, 0) U (6, ∞).