x^3-6x^2>0
anser in interval notation
To determine the interval in which the inequality x^3-6x^2>0 is true, we need to find the values of x that satisfy the inequality. Let's solve it step by step:
Step 1: Factor the expression x^3-6x^2 to rewrite it as x^2(x-6).
Step 2: Set each factor equal to zero and solve for x separately:
x^2 = 0 => x = 0
x - 6 = 0 => x = 6
Step 3: Plot the critical points on a number line:
-∞ | 0 | 6 | +∞
Step 4: Choose a test point in each interval:
Let's test x = -1 (a value less than 0):
Plugging x = -1 into the expression x^3 - 6x^2:
(-1)^3 - 6(-1)^2 = -1 - 6 = -7 (which is negative)
Let's test x = 1 (between 0 and 6):
Plugging x = 1 into the expression x^3 - 6x^2:
1^3 - 6(1)^2 = 1 - 6 = -5 (which is negative)
Let's test x = 7 (a value greater than 6):
Plugging x = 7 into the expression x^3 - 6x^2:
7^3 - 6(7)^2 = 343 - 294 = 49 (which is positive)
Step 5: Determine the intervals where the expression is greater than zero:
- From -∞ to 0, the expression is negative;
- From 0 to 6, the expression is negative;
- From 6 to +∞, the expression is positive.
Step 6: Representing the intervals in interval notation:
The solution to x^3-6x^2>0 in interval notation is (-∞, 0) U (6, +∞).