Determining Increase and Decrease of Functions | A Step-by-Step Guide with Example

To Find Where f is Increasing or Decreasing

To determine where a function is increasing or decreasing, we can find the intervals on which the function’s derivative is positive or negative

To determine where a function is increasing or decreasing, we can find the intervals on which the function’s derivative is positive or negative.

1. Start by finding the derivative of the function, denoted as f'(x).
2. Set f'(x) greater than zero, f'(x) > 0, and solve for x to find the intervals where the derivative is positive. This indicates where the function is increasing.
3. Set f'(x) less than zero, f'(x) < 0, and solve for x to find the intervals where the derivative is negative. This indicates where the function is decreasing. 4. Any points where the derivative is zero or undefined should be checked separately as they might indicate critical points, such as local maxima or minima. Here's an example to illustrate the process: Let's consider the function f(x) = x^2 - 4x + 3. Step 1: Find the derivative f'(x): f'(x) = 2x - 4 Step 2: Set f'(x) > 0 and solve for x:
2x – 4 > 0
2x > 4
x > 2

So, the function is increasing for x > 2.

Step 3: Set f'(x) < 0 and solve for x: 2x - 4 < 0 2x < 4 x < 2 Therefore, the function is decreasing for x < 2. Step 4: Check any critical points by finding where f'(x) = 0 or undefined: 2x - 4 = 0 2x = 4 x = 2 At x = 2, the derivative is zero. This indicates a possible local extremum. To determine if it's a maximum or minimum, we can check the sign of the second derivative or examine the behavior of the function around x = 2. Overall, the function f(x) = x^2 - 4x + 3 is increasing for x > 2, decreasing for x < 2, and may have a local extremum at x = 2.

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