Mathematics

Problem

Let

F_{a}(x)=x^{3}+ax

, where

a

is a real parameter. Answer the following: a. (i) Find

F_{a}'(x)

. b. (ii) Determine all real values of

a

for which

F_{a}

has no local extremum on

\mathbb R

. Give a complete justification based on the sign of

F_{a}'(x)

. c. (iii) Choose one value of

a

for which

F_{a}

does possess local extrema. Locate these points and classify each as a local maximum or a local minimum.