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


f(x) = √2 sin x- x on [0, 2π]

Verified step by step guidance

1. First, find the derivative of the function f(x) = √2 sin x - x. The derivative, f'(x), will help us determine where the function is increasing or decreasing.
2. Calculate the derivative: f'(x) = √2 cos x - 1. This involves differentiating √2 sin x to get √2 cos x and differentiating -x to get -1.
3. Set the derivative f'(x) = √2 cos x - 1 equal to zero to find the critical points. Solve the equation √2 cos x - 1 = 0 for x within the interval [0, 2π].
4. Determine the sign of f'(x) in each interval created by the critical points. This will tell us where the function is increasing (f'(x) > 0) and where it is decreasing (f'(x) < 0).
5. Finally, use the sign analysis to identify the intervals on [0, 2π] where the function f(x) is increasing and where it is decreasing. Remember that the function is increasing where f'(x) > 0 and decreasing where f'(x) < 0.

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 at any point on the function's graph. 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 evaluate the derivative at these critical points and test the sign of the derivative in the intervals created by these points.

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.