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Study Tips: Maxima and Minima

This tip on calculus was contributed by one of our top tutors, Darshan Maheshwari.

The last edition of study tips, we posted an interesting trick to differentiate an infinite series. This time we’re following it up with helpful formulae to quickly find maxima or minima for functions involving two variables.


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Consider the following problem:

a, b and c are arbitrary constants. x and y are always positive. We are also given the relation

where k is a constant.

The condition at which value of (ax + by), i.e, value of c is minimum is

Wondering how? Here is the explanation:
Given

Substitute the above value of y in the equation

So we get

Differentiate the above equation and equate to zero

Therefore,

But we already know that

So we can conclude that

But wait! How do we know this is the minima?
Simple. We differentiate the equation again.

The above equation is positive for all positive values of x. Thus we know that the value of x obtained by differentiating c and equating it to zero is the minima.

Conversely, the maximum value of i.e, the maximum value of k is also at

How did we arrive at this result? Try it for yourself! Find y in terms of x as

Now substitute this value in

Differentiate once and equate with zero to find the condition at which the maximum value is obtained.

To check if it’s the maxima, differentiate again and see if the value of is positive or negative.

Easy, right? Stay tuned for the next edition of Study Tips by HashLearn!


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