With respect to its derivative, when is f(x) decreasing?
A function f(x) is decreasing in an interval if its derivative, denoted as f'(x), is negative in that interval
A function f(x) is decreasing in an interval if its derivative, denoted as f'(x), is negative in that interval. In other words, the function is decreasing when its slopes are negative.
To determine when f(x) is decreasing, you need to follow these steps:
1. Find the derivative of f(x), i.e., f'(x).
2. Set f'(x) < 0 to find the intervals where the function is decreasing.
3. Solve for x to determine the values or range where f(x) is decreasing.
Let's walk through an example to illustrate this process. Suppose you have the function f(x) = x^2 - 4x + 3 and you want to find the intervals where it is decreasing.
1. Find the derivative of f(x):
f'(x) = 2x - 4
2. Set f'(x) < 0:
2x - 4 < 0
3. Solve for x:
2x < 4
x < 4/2
x < 2
The solution x < 2 represents the interval where f(x) is decreasing. So, for all values of x less than 2, the function f(x) is decreasing.
Remember, this is just one example, and the process may vary depending on the complexity of the function. But the key idea is to find the intervals where f'(x) is negative to determine when f(x) is decreasing.
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