Question

Let f be the function defined by f(x) = 3x^5-5x^3+2
a) On what intervals is f increasing
b) On what intervals is the graph of f concave upward.
c) Write the equation of each horizontal tangent line to the graph of f.
Please provide some detail to your response.

Answer 1

firstly, we want to know the the stationary points

Answer 2

d/dx(3 x^5-5 x^3+2) = 15 x^2 (x^2-1)

Answer 3

@ stationary point \[d/dx(3 x^5-5 x^3+2) = 15 x^2 (x^2-1)=0\]

Answer 4

\[\therefore x=0 and x=1\]

Answer 5

\[f(0)=2 and f(1)=0\]

Answer 6

a. on what interval is f increasing ?

Answer 7

a) f is increasing on the intervals at which the derivative is negative:
\[f(x)=3x^5-5x^3+2 \implies f'(x)=15x^4-15x^2=15x^2(x^2-1).\]
Do you know how to study signs of f'(x)?

Answer 8

\[x =\pm 1\]

Answer 9

It's pretty easy. \(15x^2\) is always positive, while x^2-1 is negative on (-1,1) and positive elsewhere. Thus f(x) is increasing on \((-\infty,\infty)-(-1,1)\).

Answer 10

a. it increases at -1

Answer 11

And decreasing on \([-1,1]\).

Answer 12

http://www.wolframalpha.com/input/?i=+3x%5E5-5x%5E3%2B2 check this out!

Answer 13

that's for b

Answer 14

now for c

Answer 15

this might tricky b/coz of the asymptote

Answer 16

at x=0 , y=2

Answer 17

this can be done by graph method or equation method which do you prefer ?

Answer 18

sorry i haven't been responding probably eq'n method

Answer 19

alryt, i think we shud draw the line at y=2

Answer 20

it shud be a str8 line

Answer 21

in c. we're asked to find the equation of the tangent line

Answer 22

we know our max=-1and min=1

Answer 23

plug these into the first equation

Answer 24

A plot is attached.