Question

The function f(x) is defined for x >= 0 by
f(x) = x^a - x^2a
a > 0 is a constant

How do I solve this for f(x) = 0? where do i find f(x) >0 and f(x) <0

Answer 1

is it \(x^a-x^{2a}\) ?

Answer 2

Yes

Answer 3

you might try with numbers for \(a\) and see how it looks. for example, if \(a=2\) you get \(x^2-x^4\)
that should give you a clue as to how to proceed

Answer 4

first notice that you can factor, to get \(x^2(1-x^2)\) and in general you can factor and get \(x^a(1-x^a)\)

Answer 5

How about multiplying the function with \[x ^{-2a}\]

Answer 6

then you would not have the same function any more, would you?

Answer 7

That is true

Answer 8

Okay so now that we have \[x ^{a}(1-x ^{a})\]

Answer 9

for it to equal 0, either \[x^{a} must be 0\]

Answer 10

or 1-x^a must be 0

Answer 11

correct?

Answer 12

yes i believe so.
btw is \(a\) really just some arbitrary constant, or is it a whole number?

Answer 13

That is what is confusing me as well. On the question sheet is simply says "where a > 0 is a constant"

Answer 14

not entirely sure how to interpret it

Answer 15

yeah then i am not either
if it was an integer you would have a polynomial, but otherwise you could have something with goofy radicals
\[\sqrt[3]{x}(1-\sqrt[3]{x})\] for example

Answer 16

Why \[\sqrt[3]{x}\]?

Answer 17

By the way, I would like to tell you that i appreciate you trying to help

Answer 18

i just picked \(a=\frac{1}{3}\) as an example

Answer 19

If we assume it is an integer, then would it be right to say
\[f(x) \ge 0\] when 0 < x > 1
and
\[f(x) \le 0\]
when x<0 and x>1

Answer 20

so it is not clear to me at all what the possible values of \(a\) are
perhaps \(a\) could be negative as well right? constant just means some number

Answer 21

it says a > 0

Answer 22

sorry if I didnt make that clear

Answer 23

well still it is not clear. here is what it looks like if \(a=2\)
http://www.wolframalpha.com/input/?i=x^2-x^4&dataset=
here is what it looks like if \(a=3\)
http://www.wolframalpha.com/input/?i=x^3-x^6
try some and see if you can generalize
i gotta run good luck

Answer 24

Thanks! take care

Answer 25

Oh wow the graphs really helped. If the function is defined for
x≥0
then the function is only greater than 0 when x is between 0 and 1
1>x<0