for \(\displaystyle f(x)=x^3-kx\), where \(\displaystyle k\) is an element of real (R), find the values of k such that f has

a. no critical numbers

b. one critical number

c. two critical numbers

\(\displaystyle f(x)=x^3-kx\)

\(\displaystyle \implies f'(x)=3x^2-k = 0\)

When \(\displaystyle k<0\) the expression is always \(\displaystyle >0,\) \(\displaystyle i.e.,\) never \(\displaystyle =0,\) hence no criticals.

When \(\displaystyle k=0,\) the only critical is \(\displaystyle 0.\)

When \(\displaystyle k>0,\) there are two criticals. \(\displaystyle (\pm \sqrt{\frac{k}{3}})\)