Could someone please explain the difference between minimizing and maximizing functions or give me some links to explain the difference in very very very simple terms?
I have searched online and I found the following link:
Maximizing and Minimizing a function
but I still dont undestand :-/
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$\begingroup$When we talk of maximizing or minimizing a function what we mean is what can be the maximum possible value of that function or the minimum possible value of that function.
This can be defined in terms of global range or local range.
When we say global range, we want to find out the maximum value of the function over the whole range of input over which the function can be defined (the domain of the function).
When we say local range, it means we want to find out the maximum or minimum value of the function within the given local range for that function (which will be a subset of the domain of the function).
For example: Lets take the function sin x. This has a maximum value of +1 and a minimum value of -1. This will be its global maxima and minima. Since over all the values that sin x is defined for (i.e. - infinity to +infinity) sin x will have maximum value of +1 and minimum of -1.
Now suppose we want to find maxima and minima for sin x over the interval [0,90] (degree). This is local maxima and minima because here we have restricted our interval over which the function is defined. Here sin x will have minimum at 0 and its value will be 0. Its maximum will be at 90 and value will be 1. All other values of sin x will lie between 0 and 1 within the interval [0,90].
$\endgroup$ 10 $\begingroup$In both situations you are finding critical points where the derivative is zero. This means that the function is not changing at this point, implying it has reached a maximum, minimum, or saddle point. To check which it is you can use the values at that point in the function and see if it is smaller or larger than points near it. Think about a parabola $y=x^2$. Here, you have a minimum at $x=0$ because it $y(x=0)=0$ is less than every point in the domain. If instead you had $y=-x^2$ you would have a maximum at $y(x=0)=0$ because $0$ is greater than all of the other points in the domain. For example, for $y(x)=-x^2$, it is always negative ($y(x=1)=-1$), so $0$ must be the maximum.
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