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


f(x) = x² - 2 ln x

Verified step by step guidance

1. To determine where the function \( f(x) = x^2 - 2 \ln x \) is increasing or decreasing, we first need to find its derivative, \( f'(x) \). Differentiate \( f(x) \) with respect to \( x \).
2. The derivative of \( f(x) = x^2 - 2 \ln x \) is \( f'(x) = 2x - \frac{2}{x} \). This is obtained by using the power rule for \( x^2 \) and the derivative of \( \ln x \), which is \( \frac{1}{x} \).
3. Set the derivative \( f'(x) = 2x - \frac{2}{x} \) equal to zero to find the critical points. Solve the equation \( 2x - \frac{2}{x} = 0 \) for \( x \).
4. Solve the equation \( 2x - \frac{2}{x} = 0 \) by multiplying through by \( x \) to clear the fraction, resulting in \( 2x^2 - 2 = 0 \). Factor or use the quadratic formula to find the values of \( x \).
5. Once the critical points are found, use a sign chart or test intervals around these points in \( f'(x) \) to determine where \( f'(x) > 0 \) (function is increasing) and where \( f'(x) < 0 \) (function is decreasing).

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 find intervals of increase or decrease, one must first identify these critical points and then test the sign of the derivative in the intervals they create.

Test Intervals

Test intervals are segments of the domain of a function that are determined by the critical points. By selecting test points within these intervals and evaluating the sign of the derivative, one can ascertain whether the function is increasing or decreasing in each interval. This method provides a systematic approach to understanding the overall behavior of the function across its domain.