Calc1231W9T4 - Classifying Stationary Points

Wednesday, 19 September 2018

12:22 AM

Machine generated alternative text: The differentiable function f(x) is said to have a stationary point at x = a when The nature of a stationary point depends upon when the function has the next non-zero derivative. From the Calculus notes, Corollary 42.6, we find the following criterion Classification of Stationary Points Suppose that f (1) (a) = O and n 2 1 is the smallest positive integer for which f (n) (a) # 0. Then • ifn is odd, then f has a horizontal point of inflexion at x = a • ifn is even and f(n) (a) > 0 then f has a local minimum at x = a , • ifn is even and f(n) (a) < O then f has a local maximum at x = a For example; consider the function x Either by calculating the derivatives, or reading them offthe degree 2 Taylor polynomial for x at x —o 2 2! we find that Then the smallest positive integer for which f (n) (0) # 0 is n local minimum ' O at x = 0 2 O _ By applying the Classification of Stationary Points result above, we deduce that x has a

 

Machine generated alternative text: The polynomial has a stationary point at x Calculate the higher derivatives: This is because So the smallest positive integer n > 1 for which f (n) (—1) 0 is + 14 + + 9 T, 2 4 Hence the function has a local minimum O at

 

Machine generated alternative text: The polynomial f(x) — x —4x +5x —5x + 4x —1 has a stationary point at = L This is because Calculate the higher derivatives: So the smallest positive integer n > 1 for which f (n) (1) # 0 is 5 Hence the function has a horizontal point of inflexion O at

 

Machine generated alternative text: Suppose that an unspecified function f@) has the degree 7 Taylor polynomial at x = 0 —12æ7 —13T —5X —15T . 4 P7 (x) Because k! you can read offthe value of the derivatives of f(x) at x = 0 by looking at the coefficients of at as follows: k! 1 11 In P7(x) is f(l) (0) = • the coefficient of 2 21 In P7(x) is f (2) (0) = • the coefficient of 3 31 In P7(x) is f (3) (0) • the coefficient of 4 41 In P7(x) is f(4) (0) = • the coefficient of And so without differentiating, we see that the smallest positive integer n > 1 for which f (n) (0) 0 is Hence the function has a local maximum 4 O atx=0.

 

Machine generated alternative text: The following question has been adapted from the last question of the final exam for MATHI 231 from 2014 session 2. The last question is usually harder' f(0) = o, —2 Question: Let f(x) be a function satisfying and Find the Taylor polynomial of degree 2 of f at x 0 and then find Answer: The Taylor polynomial of degree 2 of f at x = O is Near x = 0 the function f(x) is equal to P2(x) plus some remainder, that is By the Lagrange formula for the remainder we know that < 18 for O <x<l lim f(x) = (x) + R3 (x). 3! Since 3! we deduce that x for some c e [O, x]. 3

 

 

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