determine value of K

euclid2

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

Anonymous1

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. $$\displaystyle (\pm \sqrt{\frac{k}{3}})$$

euclid2