1..THE POSITIVE REAL FUNCTION SATISFIES F(X)F(Y)≤7(X-Y)2 FOR X , Y BELONG TO ITS DOMAIN ,F(3/2)=Π4 THEN
1.F(X)=√X+X3 2.F'(7)=0
3.F(X)=∫sin-1√t dt +∫ cois-1√t dt
(in 3--limit for sin-1√t is from 0 to sin2x while for second one 0 to cos2x)
-
UP 0 DOWN 0 0 1
1 Answers
341Lemma: If a differentiable function satisfies for all x,y in its domain, the relation |f(x) - f(y) | \le C |x-y|^2 for some C>0, then f(x) is a constant.
Proof: 0 \le \left| \frac{f(x+h) - f(x)}{h} \right| \le Ch
By Sandwich theorem f'(x) = 0.
Here taking logarithms on both sides we get (this is permitted since range of f is the +ve real numbers) |\ln f(x) - \ln f(y)| \le 2 (x-y)^2
So if g(x) = ln f(x) then g(x) and hence f is a constant.
So Option (2) is correct
Dunno abt Option 3
