In answer to OP's original

According to Longman Dictionary,

"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: Principal Translation why not interj informal  (expressing openness to try [sth])

Compound Forms Why not? expr (why is that not so?)

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])

According to Wordreference.com: Principal Translation why not interj informal (expressing openness to try [sth])

Compound Forms Why not? expr (why is that not so?)

In answer to OP's original

According to Longman Dictionary,

"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: Principal Translation why not interj informal (expressing openness to try [sth])

Compound Forms Why not? expr (why is that not so?)