How to mininize this?

onako · 2011-03-04 00:53:36

Given a certain function f(x), the value of x for which f(x) is minimum is found through differentiation.
Suppose a function

given the same number of elements in the summation.
So, to discover the right t for which f(t) is minimum, I would need a differentiation.
But, I'm not sure how to proceed given the exponents.
Is differentiation possible here, at all?
Thanks

Bob · 2011-03-04 01:20:12

hi onako

I think so but you may not find the answer easy to use.

if

so

The other term can be done similarly.

Bob

onako · 2011-03-04 02:29:05

Thanks.
But, how do you account for the summation? Also, you're introducing the exponent term in the solution again.
The question is to find t for which f(t) is minimum.

Bob · 2011-03-04 05:30:39

hi onako

how do you account for the summation?

I think it is inevitable that there will be a summation for this.

The expression has lots of terms and the minimum will depend on all of them together.

I did warn that you wouldn't like it.

you're introducing the exponent term in the solution again.

Well, that's a property of exponentials, isn't it.

May be a specific example where the as and bs have values would help. Do you have one?

Bob

Last edited by Bob (2011-03-04 05:31:16)

onako · 2011-03-05 00:24:07

Unfortunately, the values of a_i and b_i are different.
Would the expression depend on the largest a_i and b_i dominantly (meaning, the final min value of f(t) mostly depend on some specific a_i and b_i, compared to other a_i and b_i values)?
The function would then look like:

What would now be the value of t for min f(t)? I guess this simplifies the matter

Last edited by onako (2011-03-05 00:26:32)

Bob · 2011-03-08 03:17:00

hi onako,

Just got back from a trip.

which gives a turning point at

I've tried a few values of a and b in the graph plotter

http://www.mathsisfun.com/data/function-grapher.php

and this is always a minimum.

The graph below is for a = 1.5 and b = 2

Bob

Math Is Fun Forum

#1 2011-03-04 00:53:36

How to mininize this?

#2 2011-03-04 01:20:12

Re: How to mininize this?

#3 2011-03-04 02:29:05

Re: How to mininize this?

#4 2011-03-04 05:30:39

Re: How to mininize this?

#5 2011-03-05 00:24:07

Re: How to mininize this?

#6 2011-03-08 03:17:00

Re: How to mininize this?

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