# Find k of f(x) giving the following criteria

• February 28th 2011, 03:10 PM
youngb11
Find k of f(x) giving the following criteria
$f(x)=x^3-kx$, where $k$ can be any number.

Find the values of $k$ such that $f$ has no critical numbers, one critical number, and two critical numbers.

If anyone could help me find one of those, I'll see if I can figure out the rest.

I'm assuming so far we set the derivative to $0$, but I'm not sure exactly what to do.
• February 28th 2011, 03:24 PM
NOX Andrew
$f'(x) = 3x^2 - k = 0$

$\implies 3x^2 = k$

$\implies x^2 = \dfrac{k}{3}$

$\implies x = \pm \sqrt{\dfrac{k}{3}}$

There are no real solutions when k < 0.

There is exactly one solution when k = 0.

There are exactly two solutions when k > 0.