Question

Let f(x) = (x^2−1)/(x^4+1)

(a) At which points does the graph of the f(x) have a horizontal tangent line?

(b) Draw the graph of f(x) on MATLAB(or octave online) and identify the points for horizontal tangents on the graph.

Answer 1

a)

horizontal tangent line is at points of extremum:

$f'(x)=\frac{2x(x^4+1)-4x^3(x^2-1)}{(x^4+1)^2}=0$

$2x^5+2x-4x^5+4x^3=0$

$x(2-2x^4+4x^2)=0$

$x_1=0$

let $k=x^2$ , then:

$k^2-2k+1=0$

$k=1$

$x_2=1,x_3=-1$

b)

x=-100:100;
y=(x.^2-1)./(x.^4+1);
plot(x,y)