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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