On Which of The Following Interval Is F Decreasing Calculator

Determine where a function f is decreasing using our calculator. Learn how to analyze function intervals and understand decreasing behavior.

What is a Decreasing Function?

A function f is decreasing 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 greater than the value at x₂ (f(x₁) > f(x₂)).

In other words, as the input increases, the output decreases. This is the opposite behavior of an increasing function.

Key Point: A function can be decreasing on some intervals and increasing on others, depending on its derivative or difference quotient.

How to Find Decreasing Intervals

To determine where a function is decreasing, follow these steps:

Find the derivative of the function f'(x)
Determine where the derivative is negative (f'(x) < 0)
These intervals are where the function is decreasing

Mathematically, f is decreasing on (a, b) if f'(x) < 0 for all x in (a, b).

For functions that aren't differentiable everywhere, you can use the difference quotient or test points within intervals.

Example Calculation

Let's find where the function f(x) = -x² + 4x - 3 is decreasing.

First, find the derivative: f'(x) = -2x + 4
Set the derivative less than zero: -2x + 4 < 0
Solve for x: -2x < -4 → x > 2

Therefore, f(x) is decreasing on the interval (2, ∞).

Note: The function is increasing on (-∞, 2) and has a critical point at x = 2.

Frequently Asked Questions

How do I know if a function is decreasing?

A function is decreasing on an interval if its derivative is negative throughout that interval. You can also test points within the interval to see if the function values decrease as x increases.

Can a function be decreasing on multiple intervals?

Yes, a function can have multiple intervals where it's decreasing, especially if it has multiple critical points where the derivative changes sign.

What if the derivative is zero at some points?

If the derivative is zero at isolated points, it doesn't affect the decreasing behavior. However, if the derivative is zero over an entire interval, you'll need to analyze the function's behavior more carefully.