Question

Consider the function f(x) = x + 2/x, defined over the domain 1 ≤ x ≤ 3. Draw its graph by following the steps:
a) Find f′(x);
b) Solve f′(x) = 0
c) Find where the function is increasing and decreasing
d) Find f′′(x)
e) Find where the function is concave up and concave down
f) Find the coordinates of all points of interest

Answer 1

I already have part a and b but im stuck on c

Answer 2

The function is a line. It has positive slope. It is increasing over all intervals.

Answer 3

is there a way i can prove that mathematically?

Answer 4

Draw the graph

Answer 5

awesome thanks. i can do part d but what about part e

Answer 6

Also if the first derivative is positive, the function is increasing.

Answer 7

If the second derivative is positive, it is concave up

Answer 8

the whole function or only some of it?

Answer 9

On the interval where it is positive

Answer 10

f(x) is concave up on I iff f''(x) \>= 0$ on I.

Answer 11

and to find the coordinates in part f i would use the original function and only include the numbers that are within the domino right?

Answer 12

*domain

Answer 13

do i just make x+(2/x)=0 and find values for x?

Answer 14

Where is x + (2/x)=0 coming from?

Answer 15

the orig function

Answer 16

is that the right way to do it. i dont know how to find the intersect if its a straight line??

Answer 17

That's not the original function you posted.

Answer 18

dang typo. it was meant to be F(x)=x+(2/x)

Answer 19

i did part a and be right and the fist derivative is positive so it is increasing. the second derivative is positive also so it is concave up so what steps do i need to take for finding the coordinates of all points of intersect

Answer 20

What did you get for \(F'(x)\)?

Answer 21

1-(2/x^2)

Answer 22

ok...so on what interval is \(F\) increasing?

Answer 23

Mertsj said it was increasing over all intervals

Answer 24

that was your previous function

Answer 25

ohhhh so is it sill increasing?

Answer 26

you tell me

Answer 27

ok so if the derivative is f'(x)=1-(2/x2), then would it mean that it decreases because the x in the equation is negative? ?

Answer 28

Ive worked out hat for part b) when f'(x) = 0 , x=plus and minus the square root of 2.
Is that right?