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?
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?
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
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...
" 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 ?
Answer:
>>>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...
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.