Method to determine solution is absolute minimum/maximum value
Let's spend a little time discussing some methods for determining if our solution is in fact the absolute minimum/maximum value that we're looking for. However, we will always have to use some method for ensuring that our answer is actually that optimal value that we're after.
Method 1: Use the method utilized in Finding Absolute Extrema.
It is the method utilized in the first example above. Recall that to utilizes this method the range of possible optimal values, let's call it I, ought to have finite endpoints. Also, the function we're optimizing (once it's down to single variable) ought to be continuous on I, involving the endpoints. If these conditions are satisfied then we know that the optimal value, either the maximum or minimum based on the problem, will takes place at either the endpoints of the range or at critical point which is inside the range of possible solutions.
There are two major issues which will frequently prevent this method from being utilized however. Firstly, not every problem will in fact have a range of possible solutions which have finite endpoints at both of ends. At least we'll see one example of this as we work through the remaining examples. Also, several of the functions we'll be optimizing will not be continuous once we decrease them down to a single variable and it will prevent us from using this method.
Method 2 : Use a variant of the First Derivative Test.
In this method we also will require a range of possible optimal values, I. Though, in this case, unlike the earlier method the endpoints do not required to be finite. Also, we will have to require that the function be continuous on the interior I and we will only require the function to be continuous at the ending points if the endpoint is finite & the function in fact exists at the endpoint. We'll see many problems where the function we're optimizing doesn't in fact exist at one of the endpoints. It will not prevent this method from being utilized.
Let's assume that x = c is a critical point of the function we're attempting to optimize, f ( x ) . Already we know from the First Derivative Test that if f ′ (x) =0 instantly to the left of x = c
(that means the function is raising immediately to the left) and if f ′ ( x ) = 0 instantly to the right of x = c (that means the function is decreasing instantly to the right) then x = c will be a relative maximum for f ( x ) .
Now, it does not mean that the absolute maximum of f ( x ) will takes place at x = c . However, assume that we knew a little bit more information. Assume that actually we knew that f ′ ( x ) = 0 for all x in I such that x = c . Similarly, assume that we knew that f ′( x )= 0 for all x in I such that x = c . In this case we know that to the left of x = c , provided we stay in I certainly, the function is always rising and to the right of x = c , again staying in I, we are always falling. In this case we can say that the absolute maximum of f (x ) in I will takes place at x = c .
Likewise, if we know that to the left of x = c the function is decreasing always and to the right of x = c the function is rising always then the absolute minimum of f ( x ) in I will occur at x = c.
Method 3 : Use the second derivative.
Actually there are two ways to utilize the second derivative to help us recognize the optimal value of any function and both use the Second Derivative Test to one extent or another.
The primary way to utilize the second derivative doesn't in fact help us to recognize the optimal value. What it does do is let us to potentially exclude values & knowing it can simplify our work fairly and therefore is not a bad thing to do.
Assume that we are looking for the absolute maximum of a function and after determining the critical points we determine that we have multiple critical points. Let's also assume that we run all of them through the second derivative test and find out that some of them are actually relative minimums of the function. As we are after the absolute maximum we know that a maximum (of any kind) can't takes place at relative minimums and therefore immediately we know that we can exclude these points from further consideration. We could do a same check if we were looking for the absolute minimum. Doing this might not seem like all that great of a thing to do, however it can, on occasion, lead to a nice reduction in the amount of work that we have to do in later steps.
The second way of using the second derivative to recognize the optimal value of function is actually very similar to the second method above. Actually we will have the similar requirements for this method as we did in that method. We require an interval of possible optimal values; I & the endpoint(s) may or may not be finite. We'll also have to requiring that the function, f (x) be continuous everywhere in I except possibly at the endpoints as above.
Now, assume that x = c is a critical point and that f ′′ (c ) = 0 . The second derivative test tells us that x =c has to be a relative minimum of the function. Assume however that we also knew that f ′′ ( x ) = 0 for all x in I. In this instance we would know that the function was concave up in all of I and that would in turn mean that the absolute minimum of f ( x ) in I would in fact have to be at x = c .
Similarly if x = c is a critical point and f ′′ ( x ) = 0 for all x into I then we would know that the function was concave down in I & that absolute maximum of f (x ) in I would have to be at x = c .