Final answer:
The function f(x) = x² - 4x + 6 is decreasing on the interval (-∞, 2) and increasing on the interval (2, ∞). This was determined by finding the derivative, setting it to zero to find critical points, and testing intervals around the critical point to determine the sign of the derivative.
Step-by-step explanation:
To find the intervals in which the function f(x) = x² - 4x + 6 is increasing or decreasing, we first need to find its derivative, f'(x). The derivative of a function gives us the slope of the tangent line at any point, which corresponds to the rate of increase or decrease of the function.
The derivative of f(x) is f'(x) = 2x - 4. To determine where this function is increasing or decreasing, we set the derivative equal to zero to find the critical points: 2x - 4 = 0, which simplifies to x = 2. This is our potential turning point from increasing to decreasing, or vice versa.
We then analyze the sign of the derivative before and after the critical point. For values of x < 2, say x = 1, f'(1) = 2(1) - 4 = -2, which is negative, indicating that the function is decreasing on the interval from negative infinity to 2. For values of x > 2, say x = 3, f'(3) = 2(3) - 4 = 2, which is positive, indicating that the function is increasing on the interval from 2 to positive infinity.
Therefore, the function f(x) is decreasing on the interval (-∞, 2) and increasing on the interval (2, ∞).