Instruction

1

The notion of critical

**point**functions are closely related to the notion of its derivative at that point. Namely, point is called critical if the derivative of a function it does not exist or is zero. Critical**points**are the internal points of the domain of definition of the function.2

To determine the critical

**points**of this function, you must perform several actions: find the domain of the function, calculate derivative, find the domain of definition of the derivative function, find the**point**treatment of the derivative to zero, to prove the belonging of the points defining the original function.3

Example 1Определите critical

**points**of the function y = (x - 3)2·(x-2).4

Reinitiate the domain of the function, in this case there are no restrictions: x ∈ (-∞; +∞);Compute the derivative y’. According to the rules of differentiation of product of two functions is: y’ = ((x - 3)2)’·(x - 2) + (x - 3)2·(x - 2)’ = 2·(x - 3)·(x - 2) + (x - 3)2·1. After opening the brackets it turns out the quadratic equation: y’ = 3·x2 – 16·x + 21.

5

Find the definition of the derivative of the function: x ∈ (-∞; +∞).Solve the equation 3·x2 – 16·x + 21 = 0 to find which x the derivative becomes zero: 3·x2 – 16·x + 21 = 0.

6

D = 256 – 252 = 4x1 = (16 + 2)/6 = 3; x2 = (16 - 2)/6 = 7/3.So, the derivative becomes zero at values of x equal to 3 and 7/3.

7

Determine whether found

**point**determining the original function. Since x (-∞; +∞), then both these**points**are critical.8

Example 2Определите critical

**points**of the function y = x2 – 2/x.9

Rectiability function definition: x ∈ (-∞; 0) ∪ (0; +∞), since x is in the denominator.Calculate the derivative y’ = 2·x + 2/x2.

10

The scope of the definition of the derivative function is the same as the original: x ∈ (-∞; 0) ∪ (0; +∞).Solve the equation 2·x + 2/x2 = 0:2·x = -2/x2 → x = -1.

11

So the derivative vanishes at x = -1. Done necessary but not sufficient condition for criticality. Since x=-1 falls within the interval (-∞; 0) ∪ (0; +∞), this point is critical.