# How to find max and min value of a function ?

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Ani Asoyan on 22 May 2020
Answered: Walter Roberson on 22 May 2020
Hi ... I have a function u_g
a=2; b=2; c=1; e=0.75 ; l=0.5;
u_g = @(x, x_e, N)(-0.5*a*x.^2+b*(x-x_e)-c*(N.^l)+e*u_p(x,x_e,N));
x and N are variables, the rest of them are parameters... I want to find the value of x which will bring u_g the maximum value and corresponding max value of u_g
and I want to find the value of N which will bring minimum value to u_g and corresponding value of u_g ..how can I do it ?.. do I have to fix one of the variables?
Walter Roberson on 22 May 2020
u_p is a function?
Ani Asoyan on 22 May 2020
yes u_p is a function... sorry I changed what I wanted.
x_e is also a variable
I want to find the values of N and x which will maximize u_g function and the max value of function
and also I want to find the value of N which will maximize u_g function given the value of x (for example x=0) ..
can I do that? .. Do I have to give x_e a value ?

Abdolkarim Mohammadi on 22 May 2020
Edited: Abdolkarim Mohammadi on 22 May 2020
Finding the value of inputs that minimzes or maximizes the objective function value is an optimization problem. If your function is linear, then you run the following code and optimize your function:
[x, fval] = linprog (u_g, [], []);
If your function is unimodal and relatively smooth, then you run the following code and optimize your function:
[x, fval] = fmincon (u_g, x0, [], []);
And if the landscape of your function is unknown, i.e., you don't know whether it is linear, nonlinear, multi-modal, non-smooth, etc, then you run the following code and optimize your function:
nvars = 3;
[x, fval] = ga (u_g, nvars);
You can refer to the documentation of each solver for more information.
John D'Errico on 22 May 2020
You seem to be advocating linprog for all problems. (At least those I've seen you answer.) Note that this is NOT a linear objective, so linprog is completely useless here.
Abdolkarim Mohammadi on 22 May 2020
I just wanted to give a general idea of the optimization tools besides the nonlinear ones that are suitale for those problems.

Walter Roberson on 22 May 2020
a=2; b=2; c=1; e=0.75 ; l=0.5;
u_g = @(x, x_e, N)(-0.5*a*x.^2+b*(x-x_e)-c*(N.^l)+e*u_p(x,x_e,N));
funmin = @(xxeN) u_g(xxeN(1), xxeN(2), xxeN(3));
funmax = @(xxeN) -u_g(xxeN(1), xxeN(2), xxeN(3));
lb = [-10 -10 -10]; %adjust as appropriate
ub = [10 10 10]; %adjust as appropriate
xxeN0 = [-.1 .2 .3]; %initial guess
[best4min, fvalmin] = fmincon(funmin, xxeN0, [], [], [], [], lb, ub);
[best4max, fvalmax] = fmincon(funmax, xxeN0, [], [], [], [], lb, ub);

Cristian Garcia Milan on 22 May 2020
I think that what you want is the function
fminbnd(fun)
that finds local minimum.
If you use
fminbnd(-fun)
you will get it max.
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Cristian Garcia Milan on 22 May 2020
How about using symbolic toolbox? Then you can derivate alomg x or N and solve making equal 0
Ani Asoyan on 22 May 2020
you mean syms x? ,, how can I do that properly?