if `|f(x_1)-f(x_2)|<=(x_1-x_2)^2`Find the equation of tangent to the curve `y= f(x)` at the point `(1, 2).`

Statement -1 If f(x) and g(x) both are one one and f(g(x)) exists, then f(g(x)) is also one one. Statement -2 If f(x_(1))=f(x_(2))hArrx_(1)=x_(2) , then f(x) is one-one.

Let f(x)=cos(a_1+x)+1/2cos(a_2+x)+1/(2^2)cos(a_1+x)++1/(2^(n-1))cos(a_n+x) where a)1,a_2 a_n in Rdot If f(x_1)=f(x_2)=0,t h e n|x_2-x_1| may be equal to pi (b) 2pi (c) 3pi (d) pi/2

Let f(x)=cos(a_1+x)+1/2cos(a_2+x)+1/(2^2)cos(a_3+x)+ ........+ 1/(2^(n-1))cos(a_n+x) where a)1,a_2 a_n in Rdot If f(x_1)=f(x_2)=0,t h e n|x_2-x_1| may be equal to (a) pi (b) 2pi (c) 3pi (d) pi/2

If f(x)=log((1+x)/(1-x)),t h e n (a) f(x_1)f(x_2)=f(x_1+x_2) (b) f(x+2)-2f(x+1)+f(x)=0 (c) f(x)+f(x+1)=f(x^2+x) (d) f(x_1)+f(x_2)=f((x_1+x_2)/(1+x_1x_2))

Let x _(1) , x _(2), x _(3) be the points where f (x) = | 1-|x-4||, x in R is not differentiable then f (x_(1))+ f(x _(2)) + f (x _(3))=

Let f(x)a n dg(x) be two continuous functions defined from RvecR , such that f(x_1) > f(x_2) and g(x_1) x_2 dot Then what is the solution set of f(g(alpha^2-2alpha) > f(g(3alpha-4))

consider f(X) =(1)/(1+|x|)+(1)/(1+|x-1|) Let x_(1) and x_(2) be point wher f(x) attains local minmum and global maximum respectively .If k=f(x_(1))+f(x_(2)) then 6k-9=________.

Statement-1: If alpha and beta are real roots of the quadratic equations ax^(2) + bx + c = 0 and -ax^(2) + bx + c = 0 , then (a)/(2) x^(2) + bx + c = 0 has a real root between alpha and beta Statement-2: If f(x) is a real polynomial and x_(1), x_(2) in R such that f(x_(1)) f_(x_(2)) lt 0 , then f(x) = 0 has at leat one real root between x_(1) and x_(2) .

Consider that f : A rarr B (i) If f(x) is one-one f(x_(1)) = f(x_(2)) hArr x_(1) = x_(2) or f'(x) ge 0 or f'(x) le 0 . (ii) If f(x) is onto the range of f(x) = B. (iii) If f(x) and g(x) are inverse of each other then f(g(x)) = g(f(x)) = x. Now consider the answer of the following questions. If f(x) and g(x) are mirror image of each other through y = x and such that f(x) = e^(x) + x then the value of g'(1) is

Consider that f : A rarr B (i) If f(x) is one-one f(x_(1)) = f(x_(2)) hArr x_(1) = x_(2) or f'(x) ge 0 or f'(x) le 0 . (ii) If f(x) is onto the range of f(x) = B. (iii) If f(x) and g(x) are inverse of each other then f(g(x)) = g(f(x)) = x. Now consider the answer of the following questions. Let f(x) = (kx)/(x+1) then the value of k such that f(x) is inverse of itself is