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- For what values does x have a local minimum for this equation?for the funciton f(x) = x^3 - 75xIts local maximum is____, which occurs at _____>Its local minimum is _____, which occurs at ____.construct a suitable Liapunov function of the form ax2 + cy2, where a and c are to be determined. Then show that the critical point at the origin is of the indicated type. 1.dxdt=−x3+xy2,dydt=−2x2y−y3; asymptotically stable
- find a linearization at a suitably chosen integernear a at which the given function and its derivative are easy to evaluate. ƒ(x) = x^-1, a = 0.9Show that if the graph of a twice-differentiable function ƒ(x) has an inflection point at x = a, then the linearization of ƒ at x = a is also the quadratic approximation of ƒ at x = a. This explains why tangent lines fit so well at inflection points.Suppose that f ‘(x) is negative for x < 1 and positive for x > 1. Does f have a local minimum or maximum value at x = 1?