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

f(x) = tan⁻¹ (x/(x²+2))

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First, understand that to determine where the function f(x) is increasing or decreasing, we need to find its derivative, f'(x). This will help us identify the critical points and analyze the behavior of the function.

Calculate the derivative f'(x) using the chain rule and quotient rule. The function f(x) = tan⁻¹(u) where u = x/(x²+2). The derivative of tan⁻¹(u) with respect to x is 1/(1+u²) * du/dx.

Apply the quotient rule to find du/dx for u = x/(x²+2). The quotient rule states that if u = v/w, then du/dx = (v'w - vw')/w². Here, v = x and w = x²+2.

Simplify the expression for f'(x) using the derivative found in the previous step. This involves substituting du/dx into the derivative of tan⁻¹(u) and simplifying the expression.

Determine the intervals where f'(x) > 0 (function is increasing) and f'(x) < 0 (function is decreasing). This involves solving inequalities and considering the critical points where f'(x) = 0 or is undefined.

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주요 개념

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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.

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.f(x) = tan⁻¹ (x/(x²+2))