Final answer:
To determine where the function f(x)=x²-6x+9 is increasing or decreasing, we find the critical points and test intervals on either side. The function is decreasing when x < 3 and increasing when x > 3.
Step-by-step explanation:
In order to determine where the function f(x)=x²-6x+9 is increasing or decreasing, we need to find the critical points. The critical points are the values of x where the derivative of the function is equal to zero or undefined. To find the derivative of the function, we can use the power rule. The derivative of f(x) with respect to x is f'(x) = 2x - 6.
We set f'(x) = 0 and solve for x:
2x - 6 = 0
2x = 6
x = 3
This tells us that the function has a critical point at x = 3. Now we need to test intervals on either side of this critical point to determine where the function is increasing or decreasing.
When x < 3, we can choose a test point, such as x = 2, and plug it into the derivative. If the value is positive, the function is increasing. If the value is negative, the function is decreasing. Evaluating f'(2) = 2(2) - 6 = -2, we see that the function is decreasing when x < 3.
When x > 3, we choose another test point, such as x = 4, and plug it into the derivative. Evaluating f'(4) = 2(4) - 6 = 2, we see that the function is increasing when x > 3.
Therefore, the function f(x)=x²-6x+9 is decreasing when x < 3 and increasing when x > 3.