Question 517528: show that sqrt(c) + sqrt(c-1)=2 for some number c between 1 and 2
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website!  and  are strictly increasing functions of c, and are defined for all  . In addition, they are continuous so it follows that  is continuous and strictly increasing on [1,2]. At c=1, , and at c=2, is larger than 2. By the intermediate value theorem there must be some value c within [1,2] such that  .
Update: and are continuous because they satisfy the conditions for continuity: that they exist everywhere (on [1,2]), the limit as c goes to some number exists, and the limit is equal to the number evaluated at c.
http://en.wikipedia.org/wiki/Continuous_function
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