Can Botox Be Used to Make Lips Appear Slightly Smiling Position (Curve Upwards) Than Appear Sad Position (Curve Downwards)?

Nothing good comes from weakening the lips with BOTOX.  Generally fillers are the way to go here.  So be careful what you ask for.  Just a little too much BOTOX and you will look like you had a stroke.

Botulinum toxin, whether Botox, Dysport or Xeomin, can be used to relax the native pulling down of the DAO muscle that anchors the corner of the mouth. During its relaxation by Botox, the DAO can no longer antagonize the normal upturning actions of the levator anguli oris muscles and the sad curvature of the corner of the mouth becomes more horizontal and the patient begins to look more happy than sad.

There are risks with any treatment and you should see a Botox expert for such an off-label use of Botox.

In the hands of a skilled physician, Botox can be used to slightly upturn the angles of lips that typically angle downward. This usually happens because tight muscles "pull" the corners down, and relaxing them can allow the lip to turn more upward on the edges. However, I stress that it is VERY important to find a doctor who has experience in this area. Otherwise, you may risk being unable to form certain words, sip from a straw, pucker your lips for a kiss, etc. Research wisely.. 

I have used Botox successfully for several years to lift the corners of the mouth - "turn that frown upside down". By carefully injecting the depressor anguli oris muscle, the corners of the mouth can be gently lifted and upturned. In some cases, the addition of filler is also quite useful and so I often will combine the two treatments with nice, natural results. 

You can have Botox put in the muscle that pulls the corner of the lips down.  It is an off label advanced technique.  When done properly it can relax the corners of the mouth and soften the marionette lines.  Combined with filler in this area it is a winning combination.