Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = eˣ/(e²ᵉ + 1)

Verified step by step guidance

1. First, understand that to determine where the function f(x) is increasing or decreasing, we need to find its derivative, f'(x). The derivative will help us identify the critical points and the behavior of the function.
2. Calculate the derivative f'(x) using the quotient rule, since f(x) = eˣ/(e²ᵉ + 1) is a quotient of two functions. The quotient rule states that if you have a function h(x) = u(x)/v(x), then h'(x) = (u'(x)v(x) - u(x)v'(x))/(v(x))².
3. Apply the quotient rule: Let u(x) = eˣ and v(x) = e²ᵉ + 1. Then, u'(x) = eˣ and v'(x) = 0 because e²ᵉ + 1 is a constant. Substitute these into the quotient rule formula to find f'(x).
4. Simplify the expression for f'(x) obtained from the quotient rule. This will give you the derivative in a form that is easier to analyze for sign changes.
5. Determine the intervals where f'(x) is positive (indicating f(x) is increasing) and where f'(x) is negative (indicating f(x) is decreasing). This involves solving inequalities based on the simplified derivative expression.

Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental tool in calculus used to determine the slope of the tangent line to the curve at any point. For a function to be increasing, its derivative must be positive, while a negative derivative indicates that the function is decreasing.

Critical Points

Critical points occur where the derivative of a function is either zero or undefined. These points are essential for analyzing the behavior of the function, as they can indicate potential local maxima, minima, or points of inflection. To determine intervals of increase or decrease, one must evaluate the derivative at these critical points and test the sign of the derivative in the intervals created by these points.

First Derivative Test

The First Derivative Test is a method used to determine whether a function is increasing or decreasing on specific intervals. By examining the sign of the derivative before and after each critical point, one can conclude whether the function transitions from increasing to decreasing or vice versa. This test provides a systematic approach to identifying the intervals of increase and decrease for a given function.