Find the value of k

**arze** · August 10th 2009, 08:54 PM

Find the values of k for which $\frac{x}{(x+1)^2(x-k)}$ has one stationary value.
I attempted to differentiate the equation using the cover-up and quotient rule, but the further i go the longer and more messy the result becomes, it'll take too long to post all my workings here
First:
$y= \frac{x}{(x+1)^2(x-k)}= \frac{Ax+B}{(x+1)^2}+ \frac{C}{x-k}$
when x=k with (x-k) covered, $C=\frac{k}{(k+1)^2}$
$x=(Ax+B)(x-k)+(\frac{k}{(k+1)^2})(x^2+2x+1)$
then comparing coefficients
$A+\frac{k}{(k+1)^2}=0$
$A=-\frac{k}{(k+1)^2}$
$-kB+\frac{k}{(k+1)^2}=0$
$B=\frac{1}{(k+1)^2}$
Then i inserted these values into the equation the tried to differentiate it with the quotient rule. This is where i get stuck.
thanks

**Matt Westwood** · August 10th 2009, 09:35 PM

Not sure how it would work out, but I'd be tempted to differentiate using the product rule, the various terms being:
$x$ , $\frac 1 {\left({x+1}\right)^2}$ , $\frac 1 {x-k}$ .

Then I'd see what values of $k$ make it so that what you get has a repeated root, i.e. only one value of the resulting equation equal to zero. It will probably be a quadratic in $k$ .

Splitting into partial fractions is all very well, but I thought this was only usually done when integrating. In this case you're differentiating.

**arze** · August 10th 2009, 09:46 PM

in the book this was the method they gave. but in any case, thanks!

**arze** · August 10th 2009, 11:25 PM

I tried working it out using the product rule then equated that with 0. and tried to solve for x. Is that what you meant? because I got some values for x in terms of k in like
$\frac{k \pm \sqrt{k^2-8k}}{4}$
so i'm stuck, what am i supposed to do with this?

**songoku** · August 11th 2009, 02:48 AM

Hi arze

I think you are on the right track. I'm sure you got this equation : 2x^2 - kx + k = 0

You are supposed to find the values of k, not the values of x so don't use quadratic formula.

Instead, because the curve has one stationary value, set the discriminant = 0. If the discriminant = 0, the equation will have one root (only one value of x). Thus, the curve only has one stationary point. ^^

Thread: Find the value of k

LinkBack

Thread Tools

Display

Find the value of k

Similar Math Help Forum Discussions

Find an Equation of the Tangent Line to the parabola at the given point and find....

Given 3 points, find the area of the triangle/find the volume of the parallelepiped

How to find the find local max, min and inflection points from an integral?

Find condition that wedge may move and if it does, find acceleration?

Have dv/dx need to find dx/dt ? (have velocity w/ respect displace - find Accel?)