Proof of rolle's theorem pdf

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The proof of Rolle's Theorem is a matter of examining cases and applying the Theorem on Local Extrema, Proof of Rolle's Theorem We seek a c in (a;b) with f 0(c) = 0. That is, we wish to show that f has a horizontal tangent somewhere between a and b. Keep in mind that f (a) = f (b). Since f is continuous on the closed interval [a;b], the Extreme Value Theorem says that f has a maximum value Rolle's theorem is a special case of Lagrange's mean value theorem. Statement of Rolle's theorem is, if a function is defined on [a,b] and (i) f(x) is continuous on [a,b] (ii) f(x) is differentiable on (a,b) (iii) f(a) = f(b) Then there exists a real number [cepsilon(a,b)] such that [f'(c)=0] Rolle's mean value theorem proof: Observe Math 221 { Notes on Rolle's Theorem, The Mean Value Theorem, l'H^opital's rule, and the Taylor-Maclaurin formula 1. Two theorems Rolle's Theorem. If a function y = f(x) is di erentiable for a x b and if f(a) = f(b) = 0, then there is a number a < c < b such that f0(c) = 0. Exercise. Suppose y = f(x) is a twice di erentiable function. Suppose also that there are three di erent numbers a The result follows by applying Rolle's Theorem to g. ⁄ The mean value theorem is an important result in calculus and has some important applications relating the behaviour of f and f0. For example, if we have a property of f0 and we want to see the efiect of this property on f, we usually try to apply the mean value theorem. Let us see some Rolle's theorem can also be used in proofs of existence of zeros. And it can be used to show that a function has at most one zero on an interval. On the other hand, the intermediate value theorem can be used to show that a function has at least one zero on an interval. Together the existence of exactly one zero can be implied. 3.2 Rolle's Theorem and the Mean Value Theorem Rolle's Theorem - Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f a f b '0 then there is at least one number c in (a, b) such that fc . Examples: Find the two x-intercepts of the function f and show that f'(x) = 0 at some point between the ROLLE'S THEOREM. Example The differentiability requirement in Rolle's Theorem is critical. roots at yet there is no horizontal tangent to the graph of f over the interval ! THE MEAN-VALUE THEOREM. MOTIVATION FOR THE PROOF . VELOCITY INTERPRETATION OF THE MEAN-VALUE THEOREM is the position versus time curve for a car moving along a straight road. the right side is the average velocity of the proof of Rolle's theorem. Because f f is continuous on a compact (closed and bounded) interval I = [a,b] I = [ a, b], it attains its maximum and minimum values. In case f(a) = f(b) f ( a) = f ( b) is both the maximum and the minimum, then there is nothing more to say, for then f f is a constant function and f′≡ 0 f ′ ≡ 0 on the whole Rolle's theorem is often used in a proof technique called proof by contradiction . The procedure is as follows. Suppose you want to show that statement A is true. ouY always keep track of a pool of truths already given or obtained. First you assume that statement A is false. This could possibly imp