%% Find Asymptotes, Critical, and Inflection Points
% This example describes how to analyze a simple function to find its asymptotes, maximum, minimum, and inflection point.
%% Define a Function
% The function in this example is
% f\left(x\right)=\frac{3x^2 +6x-1}{x^2 +x-3}\ldotp
% First, create the function.

syms x
num = 3*x^2 + 6*x -1;
denom = x^2 + x - 3;
f = num/denom

% Plot the function by using fplot. The fplot function automatically shows horizontal and vertical asymptotes.

fplot(f)

%% Find Asymptotes
% To find the horizontal asymptote of f mathematically, take the limit of f as x approaches positive infinity.

limit(f,Inf)

% The limit as x approaches negative infinity is also 3. This result means the line y = 3 is a horizontal asymptote to f.
% To find the vertical asymptotes of f, set the denominator equal to 0 and solve it.

roots = solve(denom)

% roots indicates that the vertical asymptotes are the lines
% x=\frac{-1-\sqrt{13}}{2}
% and
% x=\frac{-1+\sqrt{13}}{2}.
%% Find Maximum and Minimum
% You can see from the graph that f has a local maximum between the points x = –2 and x = 0. It also has a local minimum between x = –6 and x = –2. To find the x-coordinates of the maximum and minimum, first take the derivative of f.

f1 = diff(f)

% To simplify this expression, enter the following.

f1 = simplify(f1)

% Next, set the derivative equal to 0 and solve for the critical points.

crit_pts = solve(f1)

% As the graph of f shows, the function has a local minimum at
% x_1 =\frac{-8-\sqrt{13}}{3}
% and a local maximum at
% x_1 =\frac{-8+\sqrt{13}}{3}.
% Plot the maximum and minimum of f.

fplot(f)
hold on
plot(double(crit_pts), double(subs(f,crit_pts)),'ro')
title('Maximum and Minimum of f')
text(-4.8,5.5,'Local minimum')
text(-2,4,'Local maximum')
hold off

%% Find Inflection Point
% To find the inflection point of f, set the second derivative equal to 0 and solve for this condition.

f2 = diff(f1);
inflec_pt = solve(f2,'MaxDegree',3);
double(inflec_pt)

% In this example, only the first element is a real number, so this is the only inflection point. MATLAB® does not always return the roots to an equation in the same order.
% Instead of selecting the real root by indexing into inter_pt, identify the real root by determining which roots have a zero-valued imaginary part.

idx = imag(double(inflec_pt)) == 0;
inflec_pt = inflec_pt(idx)

% Plot the inflection point. The extra argument [-9 6] in fplot extends the range of x values in the plot so that you can see the inflection point more clearly, as the figure shows.

fplot(f,[-9 6])
hold on
plot(double(inflec_pt), double(subs(f,inflec_pt)),'ro')
title('Inflection Point of f')
text(-7,1,'Inflection point')
hold off

%
% Copyright 2019 The MathWorks, Inc.