Maxima And Minima of Functions of Two, Three variables
Finding out the maximum and minimum value of a function is of immense importance to the engineering industry, researchers and especially for optimization purposes. Here we will be explaining the basic concepts and exactly how to find the maximum and minimum value of function of two variables followed by several variables.
Stationary Points, Local Minimum, Local Maximum, Global minimum and global maximum
Concept of the stationary points, local maximum and minimum as well as global maximum and minimum is needed in order to find the maximum or minimum value of a function. Stationary points are the points where derivative is zero.
For a function of two variables finding out the maximum or minimum value is easier than that of a function of 3 variables. Let us concentrate on a function of two variables x and y.
That means y=f(x)
According to the theory and graphical representation it is well established that for y to be maximum or minimum dy/dx has to be equal to Zero. By doing this we will get one or two values of x according to the function.
Now if d2y/dx2<0 for the value of x we have calculated then the function has a maximum value at that point. If it becomes greater than zero then the function possesses the minimum value at that point. And if the double derivative equals to zero, it needs further investigation with the help of graphical analysis.
In case of 3 variable function z=f(x,y) the process is little more complicated but not at all difficult. As it is a function of 3 variables., finding maxima or minima means finding the highest and lowest point of a surface whose equation has been given. Herein this case the use of partial derivation is needed.
∂z/∂x and ∂z/∂y =0 and find out the values of x and y by solving the equations
then find out ∂2z/∂x2 =r
∂2z/∂x∂y=s
∂2z/∂y2=t
Now if rt-s2>0 and r<0 the function has its maximum value at that point
If r>0 them it is minimum
rt-s2<0 then it is a saddle point
and in case rt-s2=0, it is doubtful and further investigation is needed.
