SOLUTION: show that sqrt(c) + sqrt(c-1)=2 for some number c between 1 and 2

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
RELATED QUESTIONS
Proof by induction. Imagine that we are going to prove by induction that: (1/sqrt(1))... (answered by ikleyn)

(1/sqrt(1)) + (1/sqrt(2)) + (1/sqrt(3)) + ... + (1/sqrt(n)) >= sqrt(n), for all n E Z^+ (answered by ikleyn)

find an equation of the line that bisects the obtuse angles formed by the lines with... (answered by solver91311)

Given that f(x) = sqrt (3-x) , find f(m + 2). A) sqrt (1-m) B) sqrt (m+2-x) C)... (answered by checkley77)

Value of c, given {{{sin(15) = (sqrt(6)-sqrt(2))/4}}} and {{{sin(150) = 1/2}}}:... (answered by Alan3354)

Can someone help me with this i've tried and i am not ending up with the right solutions. (answered by tutorcecilia)

Show that {{{1/(sqrt(a)+sqrt(a+1))}}}={{{sqrt(a+1)-sqrt(a)}}} Hence, deduce the value of (answered by solver91311)

Value of c, given {{{sin(15) = (sqrt(6)-sqrt(2))/4}}} and {{{sin(150) = 1/2}}}:... (answered by Alan3354)

Show that 1/sqrt 1 + 1/sqrt 2 + 1/sqrt 3...+ 1/sqrt n < 2*sqrt n for all positive... (answered by venugopalramana)