You will need

- - the function you want to examine for the presence of stationary points;
- - definition of stationary points: a stationary point of a function is the point (value) where the derivative of the first order vanishes.

Instruction

1

Using a table of derivatives and differentiation formula of the functions, it is necessary to find the derivative of the function. This step is the most difficult and responsible in performing tasks. If you make a mistake at this stage, further calculations will have no meaning.

2

Check whether derivative of the function from the argument. If you find the derivative does not depend on the argument, then there is a number (for example, f'(x) = 5), in this case, the function has no stationary points. Such a solution is possible only if the investigated function is a linear function of the first order (for example, f(x) = 5x+1). If the derivative of a function depends on the argument, then proceed to the last step.

3

Write down the equation f'(x)= 0 and solve it. The equation may not have solutions - in this case, the function's stationary points is not available. If the solution to the equation is that these are the values of the argument and will be stationary points of the function. At this stage you should inspect the solution of equation method substitution argument.

Note

When finding the derivative of the function can be difficult if the function is complex. In this case you need to use the replace function part, the intermediate argument.

Useful advice

To perform this task, you must pay special attention to the rules of differentiation.

Attention and concentration on the task will also help to deal with it - before running, make sure that you don't get distracted in the decision process.

Knowledge of the stationary points of a function greatly facilitates the construction of its graph, as it is at these points is the maximum and minimum values of the function.

Attention and concentration on the task will also help to deal with it - before running, make sure that you don't get distracted in the decision process.

Knowledge of the stationary points of a function greatly facilitates the construction of its graph, as it is at these points is the maximum and minimum values of the function.

# Advice 2 : How to find critical points

The critical point of a function is the point at which the derivative of the function turns zero. The value of the function at the critical point is called a critical value.

You will need

- Knowledge in mathematical analysis.

Instruction

1

Derivative of the function at the point is the ratio of the increment function to the increment of its argument, at aspiration of increment of argument to zero. But for the standard features there are so-called derived table, and when the differentiation of functions using various formulas, greatly simplifying this step.

2

Let given the function f(x) = x^2. To search for critical points you need to find the derivative. Using the table of derivatives we easily find that the derivative of the function f(x) is: f'(x) = 2x.

3

Next, equate the derivative to zero and solve the resulting equation. As a result, the roots of this equation are critical points of the original function f(x). Equate the derivative to zero: f'(x) = 0 or 2x = 0. Solving the resulting equation, you get that x=0. This point will be critical to the original function.

Note

Some functions does not exist a critical point or value at these points is infinite, in this case through these points are the asymptotes of the function.

Useful advice

The equation of the line has no critical points, since its derivative never becomes zero, but the direct the type: f(x) = C, where C is some constant, have infinitely many critical points.