Show that if x is real, the expression x^2-bc÷2x-b-c
has no value between b and C.
Answers
Answer:
Step-by-step explanation:
Answer:
lue between [two values]”, it’s unclear whether you mean the domain or range of the expression. However, the rational expression has at most two vertical asymptotes (as the denominator has at most 2 zeros), so the domain lacks at most two values of x, and you must be referring to the range.
Looking back at the four solutions offered, we can immediately rule two of them out; we know the expression has a horizontal asymptote (due to matching degree of numerator and denominator) at y=1 (quotient of equal leading coefficients of numerator and denominator), and two vertical asymptotes, both of which exclude the two “lies between” options.
Thus, if we can find one or more local maximum or minimum values of the expression, we should be able to narrow down to one solution. Knowing that (FG)′=F′G−FG′G2 , we want to solve:
0=(2x+34)(x2+2x−7)−(x2+34x−71)(2x+2)(x2+2x−7)2
Since we only need zeroes, we can ignore the denominator (unless it has zeros at any of the same x-values as the numerator) and solve for numerator=0; the numerator simplifies to:
0=−33x2+128x−96
Solving via your method of choice yields two zeros, x={1,3} ; substituting those values into your initial expression yields output values of 9 and 5, respectively, as extrema. Since we already established that all of the values are not constrained between any two values, there must be no values between 5 and 9.
Graphical check confirms:
4.9K viewsView Upvoters
9
Add Comment
Pratyush Kumar, Student
Answered August 13, 2016
Originally Answered: If x is real, then the expression x^2 + 34x - 71 / x^2 + 2x - 7 lies between 4 and 7 or between 5 and 9 or has no value between 4 and 7 or between 5 and 9?
Take x^2+34x-71/x^2+2x-7=y
then by transposing denominator to other side and rearrànging
(y-1)x^2+(2y-34)x-7y+71=0
As x is real
D must be positive>=0
Applying b^2-4ac>=0
4y^2+1156–136x-4(y-1)(-7y+71)>=0
32y^2–448y+114y>=0
Y^2-14y+45>=0
(y-9)(y-5)>=0
Therefore y belongs to (-infinity, 5] U [ 9,infinity)
5.6K viewsView UpvotersNot for Reproduction
23
2
2 comments from Sanghahmitra Ghosh
Promoted by OPPO Mobile India
Are curved screen phones the future?
OPPO Mobile India
Answered August 7
The shift from flat-screen panels to curved display screens has been commonly seen in a variety of products such as TV’s and laptop’s but now even the smartphone industry seems to be jumping on this bandwagon. Thus, it is safe to say that curved screens are here to stay at least in the ne
(Continue reading in feed)
Related Spaces (More Answers Below)
MATHS HOMEWORK HELP
This space is for who like to search happiness in life and for who love Maths
Follow
28.6K
Logic/puzzle reasoning.
Here you can ask/post any logical reasoning as well as tricky puzzle.
Follow
78.4K
Python coders united
This blog is all about python
Follow
7
The Math Guru
School Maths CBSE, ICSE, IB and Quantitative Aptitude Questions, Math shortcuts
Follow
58.6K
Math & Science
This space is all about Math and Science.
Follow
37.1K
View More Spaces
Kuruba Hanumanthappa
Answered December 21, 2018
7.5K viewsView Upvoters
14
1
1 comment from Nandita Mehta
Related Questions
More Answers Below
What is the least value of the expression (x^2 - 6x + 5) ÷ (x^2 + 2x + 1) if x is real. Can we solve it without the differentiation method?
If f(x)=ax2+bx+c has no real zeroes and a+b+c<0 , how do I prove that c<0 ? m factoring or quadratics.
202 views Ask Question
As x is real, so D > 0.
This is possible only when either both terms are (+)ve or both (–)ve.
(y – c) (y – b) = (+)ve, when y > c and y > b
(y – c) (y – b) = (–)ve, when y < c and y < b
Hence, the given expression has no real value between b and c.