In answer to OP's original [amended here]

> According to Longman Dictionary [reference needed], 
> 
> *"Why not" is a "spoken phrase" used to say that you agree with a suggestion.*
> 
> My question is: Can I use *Why not?* in 'formal writing' as in the
> following sentence:
> 
> We then thought we might investigate whether an increase in the
> temperature resulted in a better yield. Why not?



According to Wordreference.com:
 

**why not** *interj*	*informal* 

(expressing openness to try [sth])