How to Determine When a Function is Increasing | A Step-by-Step Guide

With respect to its derivative, when is f(x) increasing?

A function f(x) is said to be increasing in a given interval if its derivative f'(x) is greater than zero for all x-values within that interval

A function f(x) is said to be increasing in a given interval if its derivative f'(x) is greater than zero for all x-values within that interval. In other words, if the slope of the function at every point in the interval is positive, then the function is increasing in that interval.

To determine when f(x) is increasing, you need to follow these steps:

1. Find the derivative of the function f(x) by differentiating it with respect to x.
2. Set up the inequality f'(x) > 0.
3. Solve the inequality to find the values of x for which the derivative is greater than zero.
4. The interval(s) obtained from solving the inequality indicate where the function is increasing.

Note that the derivative of a function represents its rate of change. When the derivative is positive, it means the function is getting larger as x increases, indicating an increasing trend.

Here’s an example to illustrate the process:

Consider the function f(x) = x^2 – 4x + 3.

Step 1: Find the derivative of f(x):
f'(x) = 2x – 4.

Step 2: Set up the inequality:
2x – 4 > 0.

Step 3: Solve the inequality:
2x > 4 (by adding 4 to both sides)
x > 2 (by dividing both sides by 2)

Step 4: The solution to the inequality x > 2 indicates that the function f(x) is increasing when x is greater than 2.

Therefore, f(x) is increasing for all x-values greater than 2.

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