By Vakhrameev S.A.

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**Extra info for A bang-bang theorem with a finite number of switchings for nonlinear smooth control systems**

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P and = = Q and than 2 n. g(f) Prom =Q have roots at they cannot have more nor Q can be all, P this it follows that neither equal to zero at all points of an area in the plane, for in that event we could select r such that the circle would pass through that area, and P and Q would vanish number at an infinite of points on this circle. The value of Q may be written, Q = r'Ysin 7*4 + ^ From sin (n - 1) + sin (n - 2) + ^ \ < <#> this expression it is readily seen that r may be taken Q has the same sign as sin n^ on all points of the so large that where sin n

A maximum. Then the left members of the above equations aro both negative. That the right members may b both negative, for very small it is 7i, necessary not only that /'(a) should vanish as that but before, /" (a) be a negative value. values of 44. Rule for Maxima and Minima. The proof of the preceding article suggests the following rule for finding maximum 0. r) = maximum Each of its ing as makes /"(x) neyaiiw or positive. Ex. it 1. /'(/,) and /"(a-) (a;) = gives x /"( 3) is maximum. Ex. 2.

R) values of f(x) = 3) 2 y^ + and is a 3 r2 LOCATION OF THE ROOTS OF AN EQUATION b Between, two xnwwive real roots a an<\ 45. Rolle's Theorem. r) = there lien at leant one real root of the ()' equation f'(x) = 0. Let the curve in this figure be the graph of f(x) = 0. *) 0. the curve bends down and -2V N then up. Between the real root at and the double root at P the curve goes up, down, up, and finally down. Evidently, between each pair of distinct successive real roots there must be at least one maximum or minimum value of f(x).