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Do problems on Maxima and Minima haunt you? So please stop nightmaring it. And go through the article. Try to get into the thorough concept.

According to USFDA, a combination product is one composed of any combination of a drug and device; biological product and device; drug and biological product

Before starting doing sums related to maxima and minima. Let's have a brief tour over the basics of Maxima and Minima.

( I ) A function,f (x

; or drug, device, and biological product and fixed dose combination would include two or more combinations of drug.

Examples of combination products may in

) attains its maximum value at point x = c

When f (c + h ) – f (c ) < 0 ,[ where h is a very small increment to c]

And f

lude drug-coated devices, drugs packaged with delivery devices in medical kits, and drugs and devices packaged separately but intended to be used together.

(c- h ) – F ( c ) < 0
( II) And f(x ) attains its minimum value at point x = c

When f (c+ h ) – f ( c ) > 0

And f (c

here is enormous increase in the number of combination products entering the market in the recent years. Combination products have proven advantages but fixe

– h ) – f ( c ) > 0

And now the problem is how to determine whether the function attains its maximum or mininimum value at point c and a

d dose combinations are still in the process of convincing regulatory authority on their advantages over the single ingredient formulations.

Combination pro

so how to determine the value of point c. The steps involve to find out the above is described below

(1 ) 1st of all we have to find f ‘

ucts have become life saving products for the pharmaceutical companies who doesn’t have many innovative molecules in their product pipeline and have been inc

(x) and f “ (x)

[ Where f ‘(x ) = dy/ dx , and f “ ( x ) = d?x / dy? ]

( 2 ) Then we have to equate f ‘ ( x ) = 0 and have to

easingly used in the product life cycle management. Even the companies having product patents are trying to extend their product life cycle through the combi

sove out the corresponding value(s ) of x ., Let they be c1 and c 2.

(3 ) Now have to find the f “ ( c1 ) and f ( c 2 )

( 4 ) now

nation products and maximize the revenues. But the companies involved in this practice are overlooking that they are burdening the patients both economically

if f” ( c1 ) > 0 ;the function will attain the minimum value at x = c1

And if f “ ( c 2 ) < 0 ; the function will attain the maxi

and physically. They need to rightly judge the benefits of the combination products and they have to even look at the risks involved when combining the produ

mum value = c2

Now we will try to relate the above when solving problems based on maxima and minima.

(a) Find the turning point(s)

ts. Some of the combination products were well accepted by physicians while others suffered. Companies involved in development of combination products are fi

of the following function and atain it is maximum &or minimum.

Y = x3 – 9x 2 + 15x + 11

To find the turning point(s) of the given func

ding difficulty in defining their combination products and facing various challenges from selecting a combination to marketing it.

Following aspects would a

tion we have to follow the steps as shown below

Step : 1

The given function Y = f ( x) = x3 – 9x 2 + 15x + 11

1st of all We ha

dd to the challenges in developing combination products:

Which markets to tap where the combination products can do fairly well?
Which combination prod

ve to find ,dy/dx i.e . ( f ‘ (x ) ) and we also have to find f”(x)

f(x) = x3 – 9x 2 + 15x + 11

f ‘( x ) = 3 x ? - 18 x + 15

cts are meaningful and rational?
Which therapeutic categories to select?
Which Combinations can address unmet needs of the patients?
Do combin

f “ ( x ) = 6 x -18

Step:2

Now equate f’(x) = 0

f’(x) = 3 x ? - 18 x + 15 = 0
Now solve for corresponding x from th

tions increase the patient compliance?
What would be the developing cost?
How to tackle the risks encountered during combination product developmen

e above equation

3 x ? - 18 x + 15 = 0

Or, 3 ( x? - 6 x + 5 ) = 0

Or, ( x? - 6 x + 5 ) = 0 [ dividing both side

t?

As combination products don't fit into the traditional categories of drugs, medical devices, or biological products, the USFDA is in the process of devel

by 3 ]

Or , [ x ? - ( 5 + 1 ) x + 5 ] = 0

Or, [ x ? - 5 x – x + 5 ] = 0 [ using middle term factorization ]

Or , [

ping new procedures for reviewing their safety, efficacy and quality.

Professional from academic institutions, pharmaceutical industries, health care indust

x ( x – 5 ) –1 ( x -5 ) ] = 0

Or , [ ( x – 5 ) ( x – 1 )] = 0

Either,( x – 5 ) = 0 , or ( x – 1 ) = 0 [ using zero facto

y and representatives from various regulatory agencies are working out to design the regulatory requirements for manufacture and sale of combination products

r theorem]

We get , x = 5

And , x = 1

So the turning points of the given function are at ( x = 5 , x = 1)
Step

.

As there is an increasing trend of the combination products companies manufacturing such products should be able to tackle the problems involved in the de

: 3

We have f” ( x ) = 6x – 18

f” ( 5 ) = 6 * 5 – 18

= 12

So , f “ ( 5 ) > 0 , and the function attains its minimum v

elopment. They need to be wiser in analyzing the market trends and the regulatory requirements.

Companies that provide selfless information through particip

alue at x = 5

Now , f “ ( 1 ) = 6 * 1 – 18

= - 12

So f “( 1 ) < 0 , and the function attains its maximum value at x =

tion in industry events and feedback to regulatory authorities would be able to face the challenges and will be successful in developing combination products

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