Question

Little fun question! :D


Find the maximum value of \(\Large y= \frac{x^4-x^2}{x^6+2x^3-1} \) for x>1

Answer 1

?

Answer 2

derivatives?

Answer 3

Algebra sucks. .-.

Answer 4

taking the derivative will be too long :)

Answer 5

Find the maximum value of \(\Large y= \frac{x^4-x^2}{x^6+2x^3-1}\) for x>1

Answer 6

i think max value would be 1/6 !
m not sure

Answer 7

method is important :)
how u got to 1/6 ?

Answer 8

is 1/6 in options ?

Answer 9

subjective lol

Answer 10

oh!

Answer 11

okay these are your options-
a)1/6
b)3/4
c)2/5
d)1/3

Answer 12

option A

Answer 13

how?

Answer 14

i used LIMIT CONCEPT here

Answer 15

first i check lim tending to x-> 1-

Answer 16

\[\large \bf \lim_{x \rightarrow 1^{1-}}\frac{x^4-x^2}{x^6+2x^3-1}=- \infty \]

Answer 17

but we don't want this !
because we want for x>1

Answer 18

so i checked for :-
\[\large \bf \lim_{x \rightarrow 1^{+}}\frac{x^4-x^2}{x^6+2x^3-1}=\frac{1}{6}\]

Answer 19

but i want to check if this is maxima or not

Answer 20

so i put x=2, i get value less than 1/6

Answer 21

so this is local maxima

Answer 22

so 1/6 is our final answer

Answer 23

@mayankdevnani I'm curious, how did you get 1/6 from the limit approaching x=1 from the right?

Answer 24

\[\large \bf \frac{0}{0}~form\]

Answer 25

then apply L HOSPITAL rule

Answer 26

Elaborate. The limit exists and is zero...

Answer 27

The function has a root at x=1.

Answer 28

How'd you get 0/0 form?

Answer 29

Oh! Gosh
i made a super mistake

Answer 30

@Empty and @ganeshie8

Answer 31

@whpalmer4

Answer 32

1/6 is correct. Done through graphical method

Answer 33

@mayankdevnani

Answer 34

so how did you draw graph ?

Answer 35

used calculator to make a table and then observed the change

Answer 36

oh! but is bad way to solve it

Answer 37

solution is solution

Answer 38

hey,calc is not allowed in examination

Answer 39

in mine it is allowed

Answer 40

oh! ok

Answer 41

although separate marks are given for method

Answer 42

hmm

Answer 43

You can factor it, I dunno if that's helpful to you.

\[\frac{x^2 (x+1)(x-1)}{(x-1)^2(x^2+x-1)^2}\]
Then cancel a term:
\[\frac{x^2 (x+1)}{(x-1)(x^2+x-1)^2}\]

Answer 44

@Empty how this will help us ?

Answer 45

@ikram002p

Answer 46

I just said I dunno if that's helpful lol.

Answer 47

lol. it doesn't really help me

Answer 48

Sorry to hear that lol

Answer 49

That denominator is part of a modified version of the generating function for the Pell numbers \(P_n\):
\[\sum_{n\ge0}P_nt^n=-\frac{t}{t^2+2t-1}\]and setting \(t=x^3\) we get something resembling the given function,
\[\sum_{n\ge0}P_nx^{3n}=-\frac{x^3}{x^6+2x^3-1}\]
How does this help? Not sure yet, if it even does... However, we can write
\[y=\left(x-\frac{1}{x}\right)\mathcal{G}(P_n;x^3)\]where \(\mathcal{G}(a_n;t)\) is the generating function of the sequence \(a_n\) with respect to \(t\).

Answer 50

I disagree with this being a "Little fun question!" I thought there was going to be something particularly clever/simple about how to solve it.

Answer 51

@ikram002p

Answer 52

x is real ?

Answer 53

yep

Answer 54

why can't we just differentiate and take root of y' ? so we would got local values, i also think this function is not bounded so that what u mean of "max" is i can't guess if it's the same as maximizing the function.
anyway here is wolfram calculation i'm too lazy to do them :P
https://www.wolframalpha.com/input/?i=maximize+(+%7Bx%5E4-x%5E2%7D%5C%7Bx%5E6%2B2x%5E3-1%7D)

Answer 55

but differentiation is too lenghty

Answer 56

lengthy

Answer 57

no it's simple enough to do polynomial over polynomial

Answer 58

so can you do for me

Answer 59

do what ?

Answer 60

differentiation

Answer 61

oh no that is boring :P

Answer 62

There is another nice solution :)

Answer 63

\(\color{white}{\text{math.stackexchange.com/questions/1380427/maximum-value-of-expression}}\)

Answer 64

That was not worth waiting 10 days for.

@mathmath333 why did you hide the link? It's not like anyone here was still trying to solve this.

I'll make it easier, for anyone who's interested:
http://math.stackexchange.com/questions/1380427/maximum-value-of-expression

Answer 65

that was exactly my method^
sry i forgot that to post solution..

Answer 66

i thought that people r still trying to solve not to spoil them , it was not hidden as u did find that