Question

Help with this Calc question?????

Use the first derivative to find all critical points and use the second derivative to find all inflection points. Use a graph to identify each critical point as a local maximum, a local minimum, or neither.

x^5-25x^4+30.

I personally got critical points of 0 and 15, with 15 being a min and 0 being neither but I was marked wrong.

How am I wrong??

Answer 1

a couple reasons

Answer 2

\[5x^4-100x^3=4x^3(x-20)\]

Answer 3

Um, that would be 5x^3(x-20)

Answer 4

oops i mean
\[5x^3(x-20)\]

Answer 5

Right, and the x equals 20 and 0

Answer 6

I meant that I put 20 not 15. 15 was the inflection.

Answer 7

in any case \(x=0\) is a critical point as is \(x=20\)

Answer 8

at \(x=0\) you have a local max, but the max is 30, not 0

Answer 9

That's what I had. 20 being a min and 0 being neither?

Answer 10

lets go slow

Answer 11

Why is the max 30 and min 0?

Answer 12

because the derivative is \(5x^3(x-20)\) you have two critical points, namely \(0\) and\(20\) as you know
\(0\) is a zero of multiplicity 3, so the derivative crosses the x axis there

Answer 13

that means it is either a local max or min, not neither
in this case it is a local max
the max is the output, not the input
if \(x=0\) you have \(f(0)=30\) so the local max is \(30\)

Answer 14

it occurs at \(x=0\) but is IS \(30\)

Answer 15

Do you plug 0 into the original function or the function found by the second derivative?

Answer 16

i plugged it in to the original function to find the y value
that y value is the local max, not the x value

Answer 17

i didn't even find the second derivative, but you can if you like

Answer 18

It asks for the x values, not y=

Answer 19

then it is misleading you too bad

Answer 20

so go with 0 then

Answer 21

and 20 is the other x, but it's a minimum because the y value is negative, correct?

Answer 22

but that is stupid
if i say "what was the highest temperature today" you would say "81 degree" not "2 pm" when the temp was 81

Answer 23

Yes, but it's asking for a critical point.

Answer 24

ok then you have them 0 and 20

Answer 25

20 being the min yeah?

Answer 26

yeah
i mean no, but that is what you are supposed to say
another reason i detest math teachers

Answer 27

the local minimum is AT \(x=20\) but in fact is IS \(f(20)=-799,970\)

Answer 28

Yep, that's the value I got.

Answer 29

I'll try switching the answers around since there's two different spots.

Answer 30

like saying "the stock marking was highest at 3 pm" instead of saying "the stock market reached "8020"

Answer 31

No, I totally understand that, but because it's a critical point it only wants where the max or min is on the x axis. I guess it makes sense because when you're finding a max or min on a graph of a derivative, you're really strictly looking at the x-axis.

Answer 32

okay

Answer 33

Thankyou, it was a matter of just switching them around ahaha.

Answer 34

yw