Another range problem

Find the range of the function defined by f(x)=( $x^{4}$ - $\sqrt{2}$ x + 2)/( $x^{4}$ - $\sqrt{2}$ x + 1).

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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`\boxed{123}`	$\boxed{123}$

Comments

I got the lower bound. But don't know how to get the upper bound. Additionally can anyone help me to find the roots of the denominator of the given function f(x)(if they exist) and if they don't then how ?

Nishant Sharma - 8 years ago

Letting x^4 - (rt2)x + 1 = y, you get f(x) = 1 + 1/y. Some work gives you y >= 1/4, giving you a maximum of 5.

Gabriel Wong - 8 years ago

That is what I could not get. Since the denominator function is increasing and decreasing in certain intervals, so I couldn't figure out how to obtain upper bound.

Nishant Sharma - 8 years ago

Use f'(x) to find out intervals in which the function is increasing or decreasing. You don't need the denominator in that case.

Aditya Parson - 8 years ago

here nishant ; ..... i think i could help you with only the answer but not the procedure how to solve it...if you don't mind i am here posting the link from where i could find answer of this question....hope this answer(only) helps you ......

Sayan Chaudhuri - 8 years ago

That was fine. Understood. Did you see my recent post ? I mean roots of denominator ?

Nishant Sharma - 8 years ago

but i could not find it....is it in the discussion forum...?...then give me the heading that you gave...

Sayan Chaudhuri - 8 years ago

Split the function into $1+\frac{1}{(x^4-\sqrt{2}x+1)}$

Since it is easy to see that the lower bound will be greater than 1 but never equal to one. Now for upper bound we can find the local maxima.

$f'(x)=-\frac{4x^3-\sqrt{2}}{(x^4-\sqrt{2}x+1)^2}$

Equating $f'(x)=0$

We get $x=\frac{1}{\sqrt{2}}$

We can check whether it is the local maxima or not using the second derivative test(checked it).

Now substituting for $x=\frac{1}{\sqrt{2}}$ in $f(x)$ We have $f(\frac{1}{\sqrt{2}})=5$ .

So range is $(1,5]$ .

Aditya Parson - 8 years ago

That was nicely explained. Got it.

Nishant Sharma - 8 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$