1. Derivation: First, we need to calculate the derivative of the function. The derivative represents the rate of change of the function at a certain point, which is the slope of the function. If the derivative is always greater than zero in a certain interval, then the function is monotonically increasing in that interval; If the derivative is always less than zero in a certain interval, the function decreases monotonically in that interval.
2. Analyze the sign of the derivative: After obtaining the derivative, we need to analyze the sign of the derivative. Specifically, we can find points with zero derivatives, which are called the critical points of functions. Then, we can analyze the sign of the interval between the critical point and the point where the derivative does not exist. In these intervals, if the derivative is greater than zero, the function monotonically increases; If the derivative is less than zero, the function is monotonically decreasing.
3. Judging the monotonicity of the function: Through the symbolic analysis of the derivative, we can determine which intervals of the function are monotonically increasing and which intervals are monotonically decreasing. In this way, we get the monotonicity of the function.
4. Use the second derivative (optional): In some cases, if the first derivative of the function is always zero in a certain interval, then we need to use the second derivative to judge the monotonicity of the function.
Specifically, if the second derivative of the function in the interval is greater than zero, then the function is concave in the interval, that is, the function is monotonically increasing in the interval; If the second derivative of a function in an interval is less than zero, then the function is convex in the interval, that is, the function is monotonically decreasing in the interval.
It should be noted that the above methods are applicable in most cases. In some special functions or special intervals, more complex analysis methods may be needed. In practical application, the monotonicity of function is often combined with the image of function. By observing the image of the function, we can also intuitively judge the monotonicity of the function.
Generally speaking, to solve the monotonicity of functions, we need to master the calculation method of derivatives and the ability to analyze the symbols of derivatives. Through the above steps, we can accurately judge the monotonicity of the function in a certain interval, which is of great significance to mathematical analysis and the solution of practical problems.