I'm supposed to find all extremum to the function f(x)=((1+x​^2−1.6x^3+​0.6x^4)/(1​+x^4)). Been struggeling with this for hours.

I'm supposed to find all extremum to the function f(x)=((1+x^2−1.6x^3+0.6x^4)/(1+x^4)). Been struggeling with this for hours.

[James Tursa]

What have you tried so far? Is your code having errors, not producing correct answers, or what?

Answer 1

[Hannes Daepp]

I understand that you want to find the minima and maxima for this function. From a plot, you can observe that this function has two global extremum, limits at x=+/-inf, and two local max/min:

>> x=-10:0.1:10;
>> f=((1+x.^2-1.6*x.^3+0.6*x.^4)./(1+x.^4));
>> plot(x,f)

You could determine absolute min/max numerically:

>> [ma, imax] = max(fvec);
>> [mi, imin] = min(fvec);
>> disp([xvec(imax) ma])
   -1.1000    2.1176
>> disp([xvec(imin) mi])
    1.9000    0.1037

Alternatively, you can use the Symbolic Toolbox to find limits and extremum: >> syms x >> f=((1+x^2-1.6*x^3+0.6*x^4)/(1+x^4)); >> limit(f,x,-inf) ans = 3/5

Minima and maxima can be found by setting the derivative equal to zero. In this case, the output has complex roots, so some extra steps must be undertaken to determine the real roots:

>> syms x f(x)
>> f=((1+x^2-1.6*x^3+0.6*x^4)/(1+x^4));
>> limit(f,inf)
ans =
3/5
>> rts=solve(diff(f,x))
rts =
                                             0
 root(z^5 - (5*z^4)/4 - z^2 - 3*z + 5/4, z, 1)
 root(z^5 - (5*z^4)/4 - z^2 - 3*z + 5/4, z, 2)
 root(z^5 - (5*z^4)/4 - z^2 - 3*z + 5/4, z, 3)
 root(z^5 - (5*z^4)/4 - z^2 - 3*z + 5/4, z, 4)
 root(z^5 - (5*z^4)/4 - z^2 - 3*z + 5/4, z, 5)

This function has several roots beyond those at zero. To find those locations, we can use "roots" and pick out the real roots:

>> roots([1 -5/4 0 -1 -3 5/4])
ans =
   1.8824 + 0.0000i
  -1.0923 + 0.0000i
   0.0466 + 1.2870i
   0.0466 - 1.2870i
   0.3666 + 0.0000i