Problem Solver
Gyanesh Kumar
Areas Gyanesh Kumar is Knowledgeable in:
I have been teaching mathematics (classes XI & XII) for the last 12 years along with my job. I have resigned in Feb, 2015. Since then, I am doing full time teaching. I am running my own coaching institute. Also, I had prepared and appeared in the “Indian Administrative Examination” by taking subject mathematics.
Gyanesh Kumar's Problem Solving Experience:
- (Q)
Use Mean value theorem to show that for 0<x<y,
(√y-√x)< (y-x)/(2√x)
Soln: Given that 0<x<y
Let f(x) =√x
⇒f(y) =√y
f′(x) =1/(2√x)
We choose ’c’ such that x<c<y
⇒f′(c) =1/ (2√c)
Then by mean value theorem (MVT), we have
f′(c) = [f(y)-f(x)]/(y-x)
⇒1/ (2√c)=(√y-√x)/(y-x)
⇒ (√y-√x)= (y-x)/(2√c)____________________________(1)
As x<c
Now (1) becomes,
(√y-√x)< (y-x)/(2√x) Proved.