Question

If you can solve the following challenge, you are good at calculus.
Let g be a function and c a number in its domain. Suppose that
g′(c) = g′′(c) = g′′′(c) = 0 and g(4)(c) > 0.(4th derivative)
Prove that g attains a local minimum at c.

Answer 1

Before we begin, do you have any idea on how to start?

Answer 2

We can use the fact that the 4th derivative to get an idea of the shape of the 3rd derivative.

Answer 3

You may think about if g has a local minimium at c. g'(c) = 0, g''(c) > 0.

Answer 4

yes. you can do that

Answer 5

sorry the fact that the 4th derivative is positive
we know that the 3rd derivative is 0 but it crosses 0 in a certain direction

Answer 6

so when x<c is the third derivative positive or negative?

Answer 7

you need to find out yourself:)

Answer 8

I know the answer, I'm just helping you out!

Answer 9

is it positive or negative?

Answer 10

the fourth derivative is positive, and how is the fourth derivative related to the slope of the third derivative?

Answer 11

it is always positive

Answer 12

the fourth derivative is just the derivative of the third derivative, isn't it?
The third derivative is a function and the derivative of it is positive

and it's positive at c
that's all we know
so the slope at c of the third derivative is positive or negative?

Answer 13

what do you think?

Answer 14

I'm asking u here and trying to help you out, not me answering my own asked questions.

Answer 15

positive. as I have answered just now

Answer 16

yes right!
so it goes through the point (c,0) with a positive slope

Answer 17

something like that
the exact shape doesn't matter
then we can figure out from the shape of that what the second derivative looks like we know it's 0 at c
and we know that the slope on the left side is positive or negative?
of the second derivative?

Answer 18

I know the answer actually. just post this question as a challenge

Answer 19

if you know, good for you:)