topic Re: Find a minimum of the function in Mathcad

Find a minimum of the function

TT_11056782 — Sun, 12 May 2024 22:48:28 GMT

Subject to:

Minimum lies in a range [0;2*π]

Write it for MathCad15 please.

Thank you

Re: Find a minimum of the function

Werner_E — Mon, 13 May 2024 02:29:06 GMT

Looks like homework, doesn't it?

Where are your attempts in MC15?

What have you already tried?

BTW, the condition f(x)>= sine is somewhat strange.
First task should be to make a plot (either by hand or via Mathcad) which shows that you are just looking for the minimum of the part of the curve plotted thick red. Fortunately this minimum (green dot) can be found easily using the zero of the first derivative ...

The name "arctg" is quite outdated and should no longer be used according to the current standard (ISO 80000-2)

In Mathcad this function is called "atan".

Re: Find a minimum of the function

vnamboodheri — Thu, 16 May 2024 05:44:45 GMT

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Re: Find a minimum of the function

TT_11056782 — Wed, 22 May 2024 16:01:29 GMT

And how do you find the solution of it?

Re: Find a minimum of the function

Werner_E — Wed, 22 May 2024 17:28:38 GMT

@TT_11056782 wrote:

And how do you find the solution of it?

As already written - define the first derivative of the function, set it to zero and solve for x. The plot suggests that you have to look in the range from pi to 2*pi.

Fortunately the position you arrive at is within the intervals of interest. Otherwise you would have to examine the endpoints of the curve fragments of interest.

Give it a try! Set up a worksheet, define the function f, define its derivative and use the "root" function or a solve block with "find" to find the solution in the range (pi; 2 pi). Don't use the symbolic "solve" as it would not be able to find a solution.

If you experience a problem while working with Mathcad, come back here, ask your question and don't forget to attach your worksheet.

Re: Find a minimum of the function

TT_11056782 — Thu, 23 May 2024 13:23:26 GMT

Is it right? I'm sorry if everything is a mess here.

Re: Find a minimum of the function

Werner_E — Thu, 23 May 2024 13:54:47 GMT

What I meant was that you may rather use the numeric solve methods provided by Mathcad out of the box. These are solve block with "find" and the "root" function.

The root function can be used in two flavors, by providing a guess value or by providing a range. Here an example for the latter (I use a range from pi to 2 pi because the plot showed that the solution is somewhere in this interval).

If you don't want to use Mathcads own numeric iteration methods but rather want to program it yourself, you sure have to look up and study the chapter on programming in the help.

You can't type the program statements in separate regions but rather you have to enter them in the so called "programming operator" as described in more detail in the help

Furthermore there is only a limited set of programming operators

Mathcads does not know an "else" (it uses "otherwise" instead) and there also is no "endwhile".

You can simply type in the programming operators but rather must pick them from the programming toolbar (Menu: View-Toolbars-Programming) or use the appropriate keyboard shortcut.

Its well worth studying this help section and also to follow the link to "Tutorial - Programming" at the end of that section.

Here is what your program could look like (I used a much smaller value for eps to enhance accuracy):

As you see its not necessary to zip or rar the Mathcad worksheet - you can attach them directly.

Re: Find a minimum of the function

TT_11056782 — Fri, 24 May 2024 12:52:02 GMT

Thank you!

and how does the constraint affects the function here?

Re: Find a minimum of the function

Werner_E — Fri, 24 May 2024 13:09:35 GMT

@TT_11056782 wrote:

Thank you!

and how does the constraint affects the function here?

Which constraint?

And what do you mean by "here"? My approach using "root" or the program I posted, base on your attempt?

Bit ignore the constraint that f(x) should be larger than the sine function given because in the plot it can be seen that this constraint is fulfilled anyway.

My root-approach looks for a solution in the range pi to 2 pi.

The program based on your attempt start with a guess value of pi and so also finds the desired solution.

Re: Find a minimum of the function

TT_11056782 — Fri, 24 May 2024 13:03:13 GMT

I mean the whole question since it's subjected to this

Re: Find a minimum of the function

Werner_E — Fri, 24 May 2024 13:15:49 GMT

@TT_11056782 wrote:

I mean the whole question since it's subjected to this

I already edited my answer above.

After making the plot and seeing that this constrained is fulfilled anyway by the minimum we get when we set the first derivative to zero, its not further considered.

To be on the safe side you can, after finding the x.min value, make a check to see if this constrained is fulfilled.

Re: Find a minimum of the function

TT_11056782 — Fri, 24 May 2024 13:17:25 GMT

Okay. Thank you!