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|dw:1439260678254:dw|

step one
find the derivative

|dw:1439260757824:dw|

let me know when you get
\[f'(x)=1-\frac{1}{2\sqrt{1-x}}\]

I'm just having trouble trying to solve for x

ok lets back up a second

and via the chain rule
\[\frac{d}{dx}[\sqrt{f(x)}]=\frac{f'(x)}{2\sqrt{f(x)}}\]

yeah the power rule works for sure, but the derivative never changes

|dw:1439261266064:dw|So would the function look like this

yeah what you wrote
now we can do it

Do we multiple the 1 by the denominator of the other fraction?

you want to set it equal to zero and solve probably a good first step

oh kk

also notice that the derivative is strictly decreasing

So does x equal 3/4 ?

which means all that is left to do is solve
\[1-\frac{1}{2\sqrt{x-1}}=0\]

yes it does

yeah you can do that if it is not obvious

then translate to increasing decreasing for \(f\) and you are done

quick quiz
\[\frac{d}{dx}[\sqrt{1-x^2}]=?\]

so (-infinity, 3/4) is incr and decreasing from (3/4, infinity)

oh no hold the phone

increasing on \((-\infty, \frac{3}{4})\) is right

1/2*sqrt(1-x^2) *-2

1/(2*sqrt(1-x^2))*-2

close
the numerator should be \(-2x\)

and of course the two's cancel

Oh yea, was in a rush

ok lets get back to your question

\[f(x)=x+\sqrt{1-x}\]right?

is it undefined from 3/4 to infinity

don't write that is increasing on \((\frac{3}{4},\infty)\) or your teacher will think you are daft

the domain is \[(-\infty, 1]\] right?

right so we just dont include the 3/4 to infinity part, right?!

so it is only increasing on \[(\frac{3}{4},1)\]

yeah you include it , but dont to to infinity

three fourth is less than one, so there is an interval over which it increases

right!

and that's because of the domain that you were explaining before

oops is said "increase" i meant "decrease"

yea I figured!

ok
want to see a picture?

sure

cool, thanks

yw
\[\frac{d}{dx}[\sqrt{x^2+2x-1}]=?\]

1/(2*sqrt(x^2+2x-1) * 2x+2)

right, better knows as
\[\frac{x+1}{\sqrt{x^2+2x-1}}\] easy right?

yea, not bad

so no more screwing around with rational exponents when dealing with square roots!!