Taylor\'s theorem can be used to determine extrema. Indeed, suppose f: [a,b] =>

Q: Taylor's theorem can be used to determine extrema. Indeed, suppose f: [a,b] =>

In calculus, the mean value theorem states, roughly: given an arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval. More precisely, if a function f is continuous on the closed interval [a, b], where a

ID: 1892341 • Letter: T

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Taylor's theorem can be used to determine extrema. Indeed, suppose f: [a,b] => R, c an interior point of [a,b] and there is an n E N for which f, and all of its nth derivatives are continuous on a neighbourhood of c with f'(c) = f"(c) =... f^(n-1)(c)= 0 but f^(n) doesn't equal 0.

Prove the following:
i. If n is even and f^(n) (c) > 0, then f(c) is a relative minimum.
ii. If n is even and f^(n) (c) < 0, then f(c) is a relative maximum.
iii. If n is odd, then f(c) is neither a relative minimum or relative maximum.

Explanation / Answer

In calculus, the mean value theorem states, roughly: given an arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. The theorem is used to prove global statements about a function on an interval starting from local hypotheses about derivatives at points of the interval. More precisely, if a function f is continuous on the closed interval [a, b], where a

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Taylor\'s theorem problem: Let f(x; y) = cos(2x + 3y). a) Find the first, second

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Taylor\'s theorem can be used to determine extrema. Indeed, suppose f: [a,b] =>

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