The research of behavior of the function having difficult dependence on an argument is conducted by means of a derivative. On the nature of change of a derivative it is possible to find critical points and sites of growth or decrease of function.
Instruction
1. On various sites of the numerical plane the function behaves differently. When crossing ordinate axis the function changes the sign, taking place zero value. Monotonous rise can be replaced by decrease when passing function through critical points — extrema. To find extrema of function, a point of intersection with coordinate axes, sites of monotonous behavior — all these problems are solved in the analysis of behavior of a derivative.
2. Before the research of behavior of function Y = F(x) estimate area of permissible values of an argument. Take cognizance only of those values of the independent x variable at which function Y existence is possible.
3. Check whether the set function is differentiated on the considered interval of a numerical axis. Find the first derivative of the set function Y' = F'(x). If F'(x)> 0 for all values of an argument, then function Y = F(x) on this piece increases. True and a converse: if on an interval of F'(x)
For finding of extrema solve the equation of F'(x)=0. Define value of an argument x ₀ at which the first derivative of function is equal to zero. If the F (x) function exists at value x = x ₀ and Y =F is equal (x ₀), then the received point is an extremum.
To define, the found extremum is a point of a maximum or a minimum of function, calculate the second derivative F"" (x) initial function. Find value of the second derivative in a point x ₀. If F"" (x ₀)> 0, then x ₀ - a minimum point. If F"" (x ₀)
4. For finding of extrema solve the equation of F'(x)=0. Define value of an argument x ₀ at which the first derivative of function is equal to zero. If the F (x) function exists at value x = x ₀ and Y =F is equal (x ₀), then the received point is an extremum.
5. To define, the found extremum is a point of a maximum or a minimum of function, calculate the second derivative F"" (x) initial function. Find value of the second derivative in a point x ₀. If F"" (x ₀)> 0, then x ₀ - a minimum point. If F"" (x ₀)