1. If f(x) is a monotonically increasing function " x Î R, f "(x) > 0 and f -1(x) exists, then prove that å{f -1(xi)/3} < f -1({x1+x2+x3}/3), i=1,2,32. Discuss the applicability of Rolle’s theorem to f(x)=log[x2+ab/{(a+b)x}] in the interval [a,b].3. y = f(x) be a curve passing through (1,1) such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2. Form the differential equation and determine all such possible curves.4. Let f be a real-valued function defined for all real nos. x such that, for some positive constant a, the equation f(x+a) = 1/2 + Ö(f(x)-(f(x))2) holds for all x. (a) Prove that the function f is periodic. (b) For a=1, give an example of a non-constant function with the required properties.Sumthe series: 1 + 4x + 9x2 + ...If w is a root of x4=1 then Show that a + bw + cw2 + dw3 is a factor ofabcdbcdacdabdabcHence Show that the det is equal to -(a+b+c+d)(a-b+c-d){(a-c)2+(b-d)2}.Provethat if a > 0, b > 0 then for any x and y the following inequalityholds true: a.2x+b.3y+1 £ Ö(4x+9y+1)Ö(a2+b2+1)Given 6 numberswhich satisfy the relations: y2 + yz + z2 = a2 z2 + zx + x2 = b2 x2 + xy + y2 = c2 Determine the sum x+y+z in terms of a, b, c. Give geometricalinterpretation if the numbersare all positive.When 0<x<1, find the sumof the infinite series: 1/(1-x)(1-x3) + x2/(1-x3)(1-x5)+ x4/(1-x5)(1-x7) + ....10. Find all real p, q, a, b such that we have (2x-1)20 - (ax+b)20 = (x2+px+q)10 for allx.11. f(x+y) = f(x) + f(y) + 2xy - 1 " x,y. f is differentiable and f ‘(0) = cos a. Prove that f(x) > 0 " x Î R.

crazy guy , 15 Years ago

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