First, the monotonicity of the function is judged. If the function is monotonous in the definition domain, the maximum value is the maximum value and the minimum value is the minimum value.

2. Derivative method

(1), and find the derivative f' (x);

(2) Find the root of the equation f'(x)= 0;

(3) Check the sign of the left and right f'(x) values of the equation. If Zuo Zheng is negative to the right, then f(x) gets the maximum at this root; If the left is negative and the right is positive, then f(x) takes the minimum value at this root.

Pay special attention to

F'(x) Meaningless points should also be discussed. You can first find the root of f'(x)=0 and the meaningless point of f'(x), which are called suspicious points, and then judge by definition.

The solution of the extreme value of the second-order continuous partial derivative function z = f(x, y) is described as follows:

(1) Solve the equation fx(x, y) = 0, and fy(x, y) = 0 to get all real number solutions, and then get all stagnation points;

(2) For each stagnation point (x0, y0), find the values of the second-order partial derivatives a, b and c;

(3) Determine the sign of AC-B2, and judge whether f(x0, y0) is extreme value, maximum value or minimum value according to the conclusion of Theorem 2.

The necessary and sufficient conditions of the extreme value introduced above are only valid when the function is derivable at the extreme point. When the function is only differentiable at some isolated points (xi, yi) in D region, these points are certainly not the stagnation points of the function, but they may be the extreme points of the function, which should be discussed separately.