Instruction
1
On different parts of the numerical plane function behaves differently. When crossing the y-axis of the function changes sign, passing through zero value. The monotonous rise may be replaced by a decrease when passing functions through the critical point — the extremes. To find the extrema of the function, the point of intersection with the coordinate axes, the plots repetitive behavior — all of these tasks are the analysis of the behavior of the derivative.
2
Before beginning the study of the behaviour of the function Y = F(x) consider the scope of permissible argument values. Take into consideration only those values of the independent variable "x" when the existence of feature Y.
3
Check whether the given function differentiable on the interval of the numerical axis. Find the first derivative of the given function Y' = F'(x). If F'(x)>0 for all values of the argument, the function Y = F(x) on this segment is increasing. The converse is true: if the interval F'(x)<0, the phase function decreases monotonically.
4
For finding extrema solve the equation F'(x)=0. Determine the argument value x₀ at which the first derivative equal to zero. If the function F(x) exists at x=h and equal Y₀=F(x₀), the resulting point is an extremum.
5
To determine the found extremum point is a maximum or minimum of a function, compute the second derivative F"(x) the original function. Find the value of the second derivative at the point x₀. If F"(x₀ )>0, then x₀ - point minimum. If F"(x₀ )<0, x₀ is the maximum point of the function.