• Oct 21st 2005, 07:00 PM
alexis
I need help with the following.......please :-(

Show that if X2 > X1 then X2 ^3 + 2X2 + 7 > X1 ^3 + 2X1 + 7 for all X2, X1 belonging to R. Without using calculus, what does this tell us about the graph of f(x) = X^3 + 2X + 7?
• Oct 21st 2005, 10:29 PM
niva
can you show f(x2)-f(x1)>0?

if you can show for every x2>x1, we can have f(x2)>f(x1)
this means this function is monotone increasing.

if still having problem please point out.
• Oct 22nd 2005, 02:28 AM
alexis
I still don't get this I need the full explanation.... sorry
• Oct 22nd 2005, 02:51 AM
niva
alright

to show X2 ^3 + 2X2 + 7 > X1 ^3 + 2X1 + 7

( X2 ^3 + 2X2 + 7 ) - ( X1 ^3 + 2X1 + 7) <------- this is f(x2)-f(x1)

(x2^3-x1^3) + (2x2-2x1) + (7-7)

Here you have 3 terms, and x2>x1

x2^3 must be greater than x1^3, so x2^3-x1^>0
2x2 must be greater than 2x1, so 2x2-2x1>0
7-7 must be zero.

sum of two greater than 0 terms and one 0 term, must be greater than 0.

then we proved X2 ^3 + 2X2 + 7 > X1 ^3 + 2X1 + 7
• Oct 22nd 2005, 03:15 AM
alexis
Thanks it's like saying this really

X2 ^3 + 2X2 > X1 ^3 + 2X1 therefore X2 > X1 but better.

Any help in the following part of the question?

Without using calculus, using what we have just explained, what does this tell us about the graph of f(x) = X^3 + 2X + 7?

About the roots, does this prove there is a point of inflextion, normally a cubic has 3 roots here there is only one etc etc...
• Oct 22nd 2005, 04:12 AM
ticbol
" Without using calculus, using what we have just explained, what does this tell us about the graph of f(x) = X^3 + 2X + 7? "

What was explained:
If (x2)^3 +2(x2) +7 > (x1)^3 +2(x1) +7,
then x2 > x1.

What is that to the graph of f(x) = x^3 +2x +7 ?

>>>That means there is no horizontal tangent line to any point on the graph. Not even at the inflection point where x=0 and y=7, or point (0,7).

>>>That means, at the positive half of the graph, which is to the right of the y-axis, the graph goes up as x approaches infinity.
Example, if x1 = 0.1, x2 = 0.3, x3 = 0.5, .....
f(0.1) = (0.1)^3 +2(0.1) +7 = 7.201
f(0.3) = (0.3)^3 +2(0.3) +7 = 7.627
f(0.5) = (0.5)^3 +2(0.5) +7 = 8.125
etc...

>>>That means, at the negative half of the graph, which is to the left of the y-axis, the graph goes down as x approaches negative infinity.
Example, if x1 = -0.1, x2 = -0.3, x3 = -0.5, .....
f(-0.1) = (-0.1)^3 +2(-0.1) +7 = 6.799
f(-0.3) = (-0.3)^3 +2(-0.3) +7 = 6.373
f(-0.5) = (-0.5)^3 +2(-0.5) +7 = 5.875
etc...
Here, you know that -0.1 is greater than -0.3, and 6.799 is greater also than 6.373. Etc...
• Oct 22nd 2005, 04:19 AM
alexis
Thanks ever so much all of you! You are real stars!
• Oct 22nd 2005, 05:16 AM
ticbol
Some more:

>>>That means, as x goes from negative infinity to positive infinity, the graph is always going up.

>>>That means all of the tangent lines to the graph, as we go from left to right, are going up or inclined to the right. That means all tangent lines have positive slopes.

>>>That means any line segment connecting two points on the graph always has a positive slope.