7 year old daughter does not speak in public

I am a mother of six. My three youngest children (all boys) have been diagnosed with autism. My daughter who is 7 has been having issues since I could remember. She exhibits severe PICA and will not speak in any public setting. My husband took her to the science fair at school last week and even when familiar children from her class approached her, she wouldn't speak. At home she is over the top with constant talking. It even goes as far as if you are trying to tell her something, she will talk right over you. She is a seemingly unhappy child and I am at a loss. Whenever she is corrected for any reason she starts crying and it is hard to redirect her. She ALWAYS has some non-food item in her mouth. On her last report card she scored very well in every are especially math. She has an average of 100% in math since kindergarten. Cognitively she is fine. My concerns are that she doesn't socialize or communicate to people outside her family well. Her teacher even wrote me a note to ask why she will not speak up when called upon. She has also been taunting her brothers and sister. She even sprayed hair remover on her sisters head. I am not sure what is going on and things just seem to be getting worse.

[MJIthewriter]

I can relate to being ornery to my sister. I think a lot of it was what you'd expect to hear. She got me annoyed, and because I got annoyed, I decided to annoy her back. Sometimes I did so just for the heck of it.... For instance I knew she was afraid of spiders (which still baffles me that something 1000x smaller than a human can cause such an anxiety attack for a "normal" human being seems a bit odd)  So I played along. I'd find a huge spider and would chase her with it... I still find it funny thinking back.

I am so mean. :D
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Speaking in public: This is something you'll need to ask your daughter about. Likely she has an answer, but either cannot put it into words, or if she does put it into words cannot speak them out the way she has them in her head.

Sense I don't know what your environment you are in I can only give you a guess at best. I’m going to throw this out for thought:

Noise level- Does she have an easier time speaking in quieter areas than noisy ones?

If this is the case, it could be the noise around her is preventing her from "hearing" her own thoughts in her head.  The noisier the environment, the less I feel like talking. I also have a hard time hearing what people have to say to me when there is lots of noise going on.

The reason why, I’m hearing a bunch of static….your voice fades into that static. When I am talking, my voice fades into that static. As a result I speak louder than I normally would, so I can hear myself over the crowd. Now if I am hushed because I am speaking too loud, I get discouraged. Either I talk and can't hear myself or I talk loud enough to be heard, but get into trouble.  If this is true for your daughter, she could have come to the conclusion not to talk.

See if there is a way to ask her to get her point of view. Please tell her about this post and see if she relates or not. I am curious.