Question

How do we determine if a function is increasing or decresing just by testing its second derivative ?

Answer 1

Sorry I dont know I am only in 7th grade lol sorry :(

Answer 2

its np

Answer 3

HI!!!

Answer 4

you want the first and second derivative right?

Answer 5

just checking it?

Answer 6

well I know how to test the fist derivate but how about the second ?

Answer 7

can I get info by just testing the second derivative ?

Answer 8

the second derivative indicates concavity
you need that?

Answer 9

I wanna test increasing decreasing

Answer 10

if there was any chance I could derive that from concativity ?

Answer 11

it does not tell you if the function is increasing or decreasing
the first derivative does that, and your answer is correct

Answer 12

without using first derivative tests

Answer 13

you cannot

Answer 14

you can check if a critical point is a local max or min using the second derivative, but you cannot use the second derivative to see if the function is increasing or decreasing
i can give you an easy example if you like

Answer 15

think of \(f(x)=x^2\) which is decreasing on \((-\infty,0)\) and increasing on \((0,\infty)\)
the second derivative is \(f''(x)=2\)

Answer 16

can I test if a function has a rel max/min with just first derivative tests ?

Answer 17

yes, you can
you can use the second derivative, or the first derivative
the check is different

Answer 18

can you also please tell me whats the intuitive difference between a stationary point and a critical point ?

Answer 19

we can use your example and check \(\sqrt2\) if you like to see if it is a max or min using the first derivative, and check again using the second derivative

Answer 20

yes pleae

Answer 21

"we can use your example and check 2√ if you like to see if it is a max or min using the first derivative, and check again using the second derivative "

Let's do that

Answer 22

ok

Answer 23

you have decided correctly that \(f\) is increasing on \((-\sqrt2,\sqrt2)\) and then decreasing on \((\sqrt2,\infty)\)

Answer 24

in other words, the derivative is positive on the first interval and then becomes negative on the second interval
that means that since it changes sign, from positive to negative at \(\sqrt2\), \(\sqrt2\) must give a local max

Answer 25

a mental picture shows it
going up, then going down

Answer 26

is that a stationary , critical point or both ? ?

Answer 27

ok now lets answer that question

Answer 28

the distinction is a stationary point \(a\) is a number with \(f'(a)=0\)

Answer 29

whereas a critical point is either \(f'(a)=0\) or \(f'(a)\) does not exist

Answer 30

so rel max/min can only be criticals

Answer 31

so all stationary points are critical points, but no vice versa
would you like a simple example?

Answer 32

is my assumption that "all rel max/min can only be criticals and not stationary" correct ?

Answer 33

no

Answer 34

ALL stationary points are also critical points
an example may help

Answer 35

so then what's the intuition behind the dinstinction of criticals and stationary

Answer 36

on a graph

Answer 37

graphicly *

Answer 38

ok i will draw two pictures both of which have 0 as a critical point, one being stationary, the other not stationary

Answer 39

the functions i will draw are very simple
one is \(x^2\) the other \(|x|\)