Question

R The Function F(X)=X2−8x, Determine The equation

Intervals where f is increasing and decreasing (in interval notation):
Increasing:
Decreasing:

Answer 1

The function f(x) = x^2 - 8x is increasing on the interval (4, ∞) and decreasing on the interval (-∞, 4), as determined by the sign of the first derivative which is f'(x) = 2x - 8.

Step-by-step explanation:

To determine the intervals where the function f(x) = x^2 - 8x is increasing or decreasing, we need to find the first derivative of the function and analyze its critical points.

The first derivative of f(x) is f'(x) = 2x - 8. To find the critical points, we set f'(x) to zero: 2x - 8 = 0, which gives us x = 4. This is the only critical point for the function on the real number line.

We can test intervals around the critical point to determine where the function is increasing or decreasing. For x < 4, say x = 3, f'(3) = 2(3) - 8 = -2, which is negative, so the function is decreasing. For x > 4, say x = 5, f'(5) = 2(5) - 8 = 2, which is positive, meaning the function is increasing.

So the interval notation for increasing and decreasing is as follows:
Increasing: (4, ∞)
Decreasing: (-∞, 4)