A function 'f' is defined as f(x) = 2x^3 - 9x^2 - 24x + 31?

Question

a) identify the possible number of real zeros
b) identify the possible number of extreme points
c) predict the end behavior of 'f'
d) compute all real zeros
e) compute the coordinates of all extrema
f) describe the rising/falling behavior of the graph

Answer 1

Use a graphing calculator.

Answer 2

a) There can only be 3 zeroes (same as roots) by the fundamental theorem of algebra because f is of degree 3.

b) The first and second derivative tell us that there are three points of inflexion, two of which are extreme points.

c) Sorry, have no idea what "end behavior" means - it must be some American term.

d)

2x^3-9x^2-24x+31= (x-1)(2x^2-7x-31)
2x^3-9x^2-24x+31= (x-1)(x-6.0575)(x+2.5575)

So zeroes are at x=1, x=6.0575 and x= -2.5575

e) f'(x) = 6x^2-18x-24
To find coords of extrema, we set f'(x)=0, so then we have:
6x^2-18x-24=0
x^2-3x-4=0
(x+1)(x-4)=0
Thus extrema are at x=-1 and x=4.
You can calculate f(-1) and f(4) yourself. If my arithmetic is correct, f(4) = -81 and f(-1)=44.

f) The graph rises on (-infinity,-1] and [4,infinity)
the graph falls on (-1,4).

Answer 3

a)possible real roots number three as the degree of the polynomial is three and the rule is as many roots as is the degree of the polynomial

b)possible extreme points=2

c)as x increases to infinity f(x) also increases to infinity

d)2x^3-9x^2-24x+31
f(1)=2-9-24+31=0
therefore x-1 is a factor
synthetically dividing by x-1
2 -9 -24 31
0 2 -7 -31
2 -7 -31 0
so the quotient on division
=2x^2-7x-31
roots=[7+/-(49+248)^1/2]/4
the real zeros are 1,[7+/-(297)^1/2]/4

e)f'(x)=6x^2-18x-24=0 =>x^2-3x-4=0 =>(x-4)(x+1)=0 so the extrema are at x=4 and x=-1
minimum at 4 and maximum at -1,the minimum and the maximum values being -81 and 45

f)graphing the curve will help us to seethe rising and falling behaviour of the curve.
however from minus infinity to -1 it rises and at -1 it changes direction.from -1 to 4 it falls.it changes direction once again at 4 and rises.at -1 it is concave downwards ( a local maximum) and at 4 it is concave upwards ( a local minimum)

Answer 4

a) 3 possible because largest power in function = x^3
b) 2 = 3-1 because largest power in function = x^3
c) as x --> infinity f(x) ---> infinity
d) solve 0 = 2x^3 - 9x^2 - 24x +31
e) solve 0 = 6x^2 - 18x - 24 (derivitive)
f) make a picture

Answer 5

I would, but I only have a couple of minutes. Just get the derivative and factor it out and you'll be on your way.

Answer 6

identify the g) question on your homework you should really do on you own.

Answer 7

ooh wow beginning calculus. i bet once you learn the rest, you'll be asking much more difficult questions

Answer 8

When in class .... pay attention...when on yahoo answers.. don't try to get your homework done..

Answer 9

hehe! i'll applaud u for trying to get ur homework done for u!

Answer 10

have no clue.... LOL... should try paying attention in school and you would not have to ask for help...