f(x)=x/(x^2+1) Does the equation f (x) - (x-1) / x = 0 have a root? If there is a root x0, ask for an interval (a, b) of length 1 / 4, so that XO ∈ (a, b). If not, explain the reason?

f(x)=x/(x^2+1) Does the equation f (x) - (x-1) / x = 0 have a root? If there is a root x0, ask for an interval (a, b) of length 1 / 4, so that XO ∈ (a, b). If not, explain the reason?

x/(x2+1)=(x-1)/x
x3-2x2+x-1=0
Let y = x3-2x2 + X-1
y=x2(x-2)+x-2+1=(x-2)(x2+1)+1
If x increases, y also increases, so y is a simple increasing function. There is an intersection between the image and the X axis, that is, f (x) - (x-1) / x = 0 has roots
Finding roots by dichotomy
x=1,y=-1
x=2,y=5
x=3/2,y=0.875
x=1.25,y=-0.296875
x=1.375,y=0.224609375
So x0 ∈ (1.25,1.375)