Critical points are points such that so you want

.

Now the only time that a fraction is equal to zero is when the top (numerator) is zero.

Therefore we want

It is difficult to see when a sum is zero but easy to see when a product is zero by the No-Zero-Divisors-Theorem which says that if and are two numbers and then either or . So we want to write the above as a product… we must factorise. What is common to both… the and :

or

or

or

These are the critical points. They might be maxima or minima (or neither — won’t happen on your paper). To do this we test them at the second derivative which is the derivative of the first derivative:

.

Now evaluate and .

If is a critical point then

implies that there is a local minimum at .

implies that there is a local maximum at .

Regards,

J.P.

I used the quotient rule to find to find the critical points and this gave me

What do I need to do next?

]]>Jim,

I presume this is it:

I have to say I like this method — moreover it works for the sum rule also.

Thank you,

J.P.

Jim,

Far easier?? Care to elaborate? Surely the difference is just cosmetic?

J.P.

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