Question

(x)=x4+4x3+100
Find the global minimum value of the function f, that is, give the minimum value of f and the value of x for which this occurs.

This is not a homework problem. I have a exam and I don't know how to answer this question. I did the first and second derivative test. I don't need answer. I don't wanna fail my exam. Can anyone explain step by step how they would do these kind of problems. I am sorry if this is a lot to explain, but I looked on youtube and all of the videos explained with the intervals. It did not help. Thank you.

Answer 1

Let's try doing it again :D

Answer 2

Could you differentiate it?

Answer 3

yeah

Answer 4

I actually took first and second derivative

Answer 5

What did you get for the first derivative?

Answer 6

x=0 and x=-3

Answer 7

What about second derivative?

Answer 8

x=0 and x=-2

Answer 9

You equated the second derivative to zero? :P

Answer 10

yeah

Answer 11

Why, though? Just get the second derivative, and I'll show you how to use it :)

Answer 12

that's how i was taught to get second derivative. ok, I will look for another way and get the value.

Answer 13

Oh, I'm sure you got the second derivative correctly, I was just asking for it in its raw form... you know, as a function of x? lol

f''(x)

Answer 14

f''(x)=12x^2+24x

Answer 15

That's better. If the first derivative gives you the rate of change (increasing, decreasing, stagnant, etc)

What does the second derivative tell you?

Answer 16

concavity?

Answer 17

Exactly. So, you know that since this is a differentiable function, its extrema could only ever be at points where the first derivative is zero, yes? :D

Answer 18

i did not know that lol

Answer 19

Oh. Well, now you do ^^

Answer 20

thank you

Answer 21

So, x = 0 and x = 3 are the values which zero out the first derivative, right?

Answer 22

Oops, I meant x = -3 lol

Answer 23

yeah

Answer 24

So, these are POSSIBLE extrema. How to find out if they are?
Take the second derivative :)

What I want you to do now is plug in x = 0 and x = -3 in the function f''(x) and tell me what you get :)

Answer 25

36 and 0

Answer 26

Okay. You know what those numbers mean, right?
lol
Positive second derivative means it's concave upward at that poinit
Zero second derivative means it's a POSSIBLE point of inflection at that point.

What's a characteristic of a minimum? :p

Answer 27

decreasing and increasing?

Answer 28

You could use that, but that's harder, and doesn't involve the second derivative test at all :D

Here's a rough sketch

Answer 29

Here's a curve with a minimum

Answer 30

i knew that ummm

Answer 31

So, what can you say about its first derivative at this point?

Answer 32

It's zero, because it's flat, right?

Answer 33

yeah

Answer 34

What about its concavity?
Does it curve up or down?