On What Interval Is F Increasing Calculator

Determining where a function is increasing is a fundamental calculus concept. This calculator helps you find the intervals where a function f(x) is increasing by analyzing its derivative.

What Is an Increasing Interval?

A function f(x) is increasing on an interval if, for any two points x₁ and x₂ in that interval where x₁ < x₂, the value of the function at x₁ is less than the value at x₂ (f(x₁) < f(x₂)).

Mathematically, a function is increasing on an interval if its derivative f'(x) is positive for all x in that interval.

How to Find Increasing Intervals

To find where a function is increasing:

Find the derivative f'(x) of the function.
Determine where f'(x) > 0.
Identify the intervals where this inequality holds true.

Note: Critical points (where f'(x) = 0 or is undefined) divide the domain into test intervals. You must test each interval to determine where the derivative is positive.

Example Calculation

Consider the function f(x) = x³ - 3x² + 4.

1. Find the derivative: f'(x) = 3x² - 6x.

2. Set f'(x) > 0: 3x² - 6x > 0.

3. Solve the inequality: x(x - 2) > 0.

The solution is x < 0 or x > 2.

Therefore, f(x) is increasing on the intervals (-∞, 0) and (2, ∞).

FAQ

What if the derivative is zero on an interval?

If the derivative is zero on an interval, the function is not increasing there. The derivative must be strictly positive for the function to be increasing.

Can a function be increasing on multiple intervals?

Yes, a function can be increasing on multiple separate intervals. For example, f(x) = x³ is increasing on (-∞, ∞) because its derivative is always positive.

What if the derivative is undefined at a point?

If the derivative is undefined at a point, that point is a critical point and divides the domain into test intervals. You must test each interval separately.