# Determining the roots, maxima, and minima of a discontinuous symbolic function

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Aleem Andrew on 27 Oct 2020
Commented: Aleem Andrew on 30 Oct 2020
How can the roots of the following discontinuity function be determined?
syms x
f =(x^3*heaviside(x))/18 - (3*x^2*heaviside(x))/2 + (27*heaviside(x - 6)*(x - 6))/4 + (27*x*heaviside(x))/4 + heaviside(x - 9)*(x - 9)^2 - (heaviside(x - 9)*(x - 9)^3)/18
fplot(f,[-2,12])
I have tried using feval and the roots functions but neither of these methods worked. I would appreciate any help.

Shubham Rawat on 29 Oct 2020
Hi Aleem,
Here is a function in syms called solve, which can evaluate roots of a function but for continous part only,
syms x
f =(x^3*heaviside(x))/18 - (3*x^2*heaviside(x))/2 + (27*heaviside(x - 6)*(x - 6))/4 + (27*x*heaviside(x))/4 + heaviside(x - 9)*(x - 9)^2 - (heaviside(x - 9)*(x - 9)^3)/18
solve(f == 0, x, 'MaxDegree', 4); % for solving the equation
roots = vpa(ans,6);
fplot(f,[-2, 12])
hold on
plot(roots, subs(f,roots), '*')
hold off
For Maxima and Minima,
syms x
f =(x^3*heaviside(x))/18 - (3*x^2*heaviside(x))/2 + (27*heaviside(x - 6)*(x - 6))/4 + (27*x*heaviside(x))/4 + heaviside(x - 9)*(x - 9)^2 - (heaviside(x - 9)*(x - 9)^3)/18
g = diff(f,x); % for differentiation
solve(g == 0,x,'MaxDegree',4); %for solving g
extrema = vpa(ans,6)
fplot(f,[-2, 12])
hold on
plot(extrema, subs(f,extrema), '*')
hold off
This code works fine for the differentiable part, for non-differentiable part you may iterate the points one-by-one how function is behaving.
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Aleem Andrew on 30 Oct 2020