Please solve all 3 parts in detail.
Part (a): f(c) = c, solve for equilibrium points, then choose different values of r for r>0.
Part (b): Find the derivative of f(x), then sub in equilibrium points to find if they are stable or unstable based on r.
Part (c): Using part (a) and (b) sketch bifurcation diagram. Use dashed lines for unstable and full lines for stable. Horizontal axis is r and veritical axis is c. Name the type of bifurcation diagram (either fold, pitchfork or transcritial).
6. Let the function f be defined by TI f (x) = 1+x¹ for z R, where r is a real positive parameter. (a) Determine the value r at which a bifurcation in the number of equilibrium points of f occurs. (b) Identify all positive values of r at which there is a change in stability of the equilibrium points corresponding to the value r. For all other positive values of r, classify the equilibrium points as stable or unstable. (c) Draw a bifurcation diagram, plotting the fixed points as a function of r and indicating their stability or instability. What is the name associated with this type of bifurcation?