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.

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Alexader Svensson
Alexader Svensson on 7 Nov 2016
Answered: Hannes Daepp on 11 Nov 2016
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.

Answers (1)

Hannes Daepp
Hannes Daepp on 11 Nov 2016
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

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