can you please solve and elaborately explain why each case is true? -specificall

Question

can you please solve and elaborately explain why each case is true?

-specifically for ii why do we only show the case for local min and inflection but doesn't draw the graph for local max, or the inflection in the other way?

-on iv how does it help us to know that the third derivative is greater than zero, can you further elaborate on this part.

August 2015 Page 1 of 11 MATH S 3027 1. Consider the autonomous differential equation y f(v) a suppose that yo is a critical point that is f(yo) the equilibrium solution (t) vo (i) 10 Indicate the asymptotic stability of f'(yo) 0. Give an example determine the asymptotic stability of the +10) Does the assumption f (yo) o equilibrium solution plt) to? Give examples. say now? Suppose f(vo) 0 and fM(yo) 0. What can you can you say now? (+5) sup (V) f (yo) but f (go) >0. What f "ly (yo) o Locat mm (yo) Sta. eg (it) f (yo local urn off atyo mafly) lautade to m fly) semistable

Explanation / Answer

(ii) for this part we can easily say that there is no need of local max graph because maxima has no information about stability. we can easily find stability by seeing graph direction and local min.

(iv)all derivative shows the slope of graph wheather it is positive or negative. thats why it helps to find stability.

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