Let `f'(x)gt0andf''(x)gt0` where `x_(1)ltx_(2).`
Then show `f((x_(1)+x_(2))/(2))lt(f(x_(1))+(x_(2)))/(2).`

Let g(x)gt0andf'(x)lt0,AAx in R, then show g(f(x+1))ltg(f(x-1)) f(g(x+1))ltf(g(x-1))

If f(x) is monotonically increasing function for all x in R, such that f''(x)gt0andf^(-1)(x) exists, then prove that (f^(-1)(x_(1))+f^(-1)(x_(2))+f^(-1)(x_(3)))/(3)lt((f^(-1)(x_(1)+x_(2)+x_3))/(3))

If the equation x^(3)-3x+1=0 has three roos x_(1),x_(2),x_(3) where x_(1)ltx_(2)ltx_(3) then the value of ({x_(1)}+{x_(2)}+{x_(3)}) is ({.} F.P. of x ).

Let f_(1) (x) and f_(2) (x) be twice differentiable functions where F(x)= f_(1) (x) + f_(2) (x) and G(x) = f_(1)(x) - f_(2)(x), AA x in R, f_(1) (0) = 2 and f_(2)(0)=1. "If" f'_(1)(x) = f_(2) (x) and f'_(2) (x) = f_(1) (x) , AA x in R then the number of solutions of the equation (F(x))^(2) =(9x^(4))/(G(x)) is...... .

Let F_(1)(x_(1),0)and F_(2)(x_(2),0)" for "x_(1)lt0 and x_(2)gt0 the foci of the ellipse (x^(2))/(9)+(y^(2))/(8)=1 . Suppose a parabola having vertex at the origin and focus at F_(2) intersects the ellipse at point M in the first quadrant and at a point N in the fourth quardant. The orthocentre of the triangle F_(1)MN , is

For x in R , x ne0, 1, let f_(0)(x)=(1)/(1-x) and f_(n+1)(x)=f_(0)(f_(n)(x)),n=0,1,2….. Then the value of f_(100)(3)+f_(1)((2)/(3))+f_(2)((3)/(2)) is equal to

Let f(x)=x^(3)+ax^(2)+bx+c be the given cubic polynomial and f(x)=0 be the corresponding cubic equation, where a, b, c in R. Now, f'(x)=3x^(2)+2ax+b Let D=4a^(2)-12b=4(a^(2)-3b) be the discriminant of the equation f'(x)=0 . If D=4(a^(2)-3b)gt0 and f(x_(1)).f(x_(2))gt0 where x_(1),x_(2) are the roots of f'(x), then

Let f(x+(1)/(x))=x^(2)+(1)/(x^(2)),(x ne 0) then f(x) equals

Statement-1 : The function f defined as f(x) = a^(x) satisfies the inequality f(x_(1)) lt f(x_(2)) for x_(1) gt x_(2) when 0 lt a lt 1 . and Statement-2 : The function f defined as f(x) = a^(x) satisfies the inequality f(x_(1)) lt f(x_(2)) for x_(1) lt x_(2) when a gt 1 .

Let f and g be differentiable functions such that: xg(f(x))f\'(g(x))g\'(x)=f(g(x))g\'(f(x))f\'(x) AA x in R Also, f(x)gt0 and g(x)gt0 AA x in R int_0^xf(g(t))dt=1-e^(-2x)/2, AA x in R and g(f(0))=1, h(x)=g(f(x))/f(g(x)) AA x in R Now answer the question: f(g(0))+g(f(0))= (A) 1 (B) 2 (C) 3 (D) 4