Question

Calculus

f(x) = ( 20)/(x^2 + 21 )

(1)Find f'(x) =

(2)Find f''(x) =

(3)Find the open interval(s) where the function is increasing:

(4)Find the open interval(s) where the function is decreasing:

(5)Find the open interval(s) where the function is concave up:

(6)Find the open interval(s) where the function is concave down:

Answer 1

what have u tried so far

Answer 2

I found F' and F''
After that I lose myself with the open intervals, I missed out on class those two days. I've been looking in the calculus book and I have no clue where to start, or which principle to use here.

Answer 3

thats a good start, so what do u have for f'(x) and f''(x) ?

Answer 4

use this for parts 3 and 4 :
f'(x) > 0 : function is increasing
f'(x) < 0 : function is decreasing

Answer 5

you just need first derivative for those parts

Answer 6

f'(x)= -(40 x)/(x^2+21)^2
f''(x)=(120 (x^2-7))/(x^2+21)^3
Do I use f' or f''?

Answer 7

f'

Answer 8

just plug a zero into f'(x) then? or a -1 and 1?

Answer 9

\(\large f'(x) = \dfrac{-40x}{(x^2+21)^2}\)

notice that the denominator is always positive since it is a square of something

Answer 10

the numerator is negative if x is positive
and is positive if x is negative

right ?

Answer 11

Yes

Answer 12

that means f'(x) > 0 : when x is less than 0
and f'(x) < 0 : when x is greater than 0

Answer 13

so can we say :

the function is increasing in interval (-infty, 0)
the function is decreasing in (0, infty)

?

Answer 14

Yes, we can, now my question is how do we know that the function is increasing in interval (-infty, 0) and the function is decreasing in (0, infty) from what is given? Does this come from a specific principle?

Answer 15

recall that the first derivative at a point represents the slope of a tangent line at that point

Answer 16

notice that the graph is increasing for the left half part
and decreasing afterwards

Answer 17

lets draw few tangents

Answer 18

Sounds good to me

Answer 19

clearly the tangents are rising when the graph is increasing, so :
\[\large \text{slope of tangents for left part is positive}\]

Answer 20

for the right its negative

Answer 21

you know that slope of tangent at a point is same as the first derivative

Answer 22

yep you get the idea

Answer 23

Ok Got it! which means that the answer to question 5 would be DNE correct?