# Finding K such that this function has critical points

• Mar 10th 2011, 03:15 PM
youngb11
Finding K such that this function has critical points
Determine the conditions on parameter $k$ such that the function $\displaystyle f(x)=\frac{2x+4}{x^2-k^2}$ will have critical points.

Critical points are defined as where $f'(x)=0$ or when $f'(x)$ does not exist. I was thinking it would be $k>0$ but the back of the book has the answer as $k$ being between $-2$ and $2$.

Was my reasoning correct or am I missing something? I'm not sure why $k$ has to be between $-2$ and $2$.
• Mar 10th 2011, 03:25 PM
skeeter
Quote:

Originally Posted by youngb11
Determine the conditions on parameter $k$ such that the function $\displaystyle f(x)=\frac{2x+4}{x^2-k^2}$ will have critical points.

Critical points are defined as where $f'(x)=0$ or when $f'(x)$ does not exist. I was thinking it would be $k>0$ but the back of the book has the answer as $k$ being between $-2$ and $2$.

Was my reasoning correct or am I missing something? I'm not sure why $k$ has to be between $-2$ and $2$.

what did you get for a derivative?
• Mar 10th 2011, 03:46 PM
youngb11
Quote:

Originally Posted by skeeter
what did you get for a derivative?

Is this correct: $\displaystyle f'(x)=\frac{-2(x^2+4x+k^2)}{(x^2-k^2)^2}$ ?

Just looking at the denominator, wouldn't a positive k value produce points where the slope doesn't exist? Unless they aren't including asymptotes. In that case, how would I figure out where the slope equaled zero?
• Mar 10th 2011, 03:59 PM
skeeter
Quote:

Originally Posted by youngb11
Is this correct: $\displaystyle f'(x)=\frac{-2(x^2+4x+k^2)}{(x^2-k^2)^2}$ ?

Just looking at the denominator, wouldn't a positive k value produce points where the slope doesn't exist? Unless they aren't including asymptotes. In that case, how would I figure out where the slope equaled zero?

the derivative is correct.

note that critical values occur at x-values where the function is defined. $f(x)$ is undefined when $x = \pm k$ , so there are no values of $x$ where $f(x)$ is defined and $f'(x)$ is undefined.

that leaves the critical values where $f'(x) = 0$

$x^2 + 4x + k^2 = 0$

$x = \dfrac{-4 \pm \sqrt{4^2 - 4k^2}}{2}$
for critical values to exist, the discriminant, $(b^2-4ac) \ge 0$
$4^2 - 4k^2 \ge 0$
solve this inequality for $k$