f is a continous function in `[a, b]`; g is a continuous function in [b,c]. A function h(x) is defined as `h(x)=f(x) for x in [a,b) , g(x) for x in (b,c]` if f(b) =g(b) then

If a twice differentiable function f(x) on (a,b) and continuous on [a, b] is such that f''(x)lt0 for all x in (a,b) then for any c in (a,b),(f(c)-f(a))/(f(b)-f(c))gt

Let S denotes the set consisting of four functions and S = { [x], sin^(-1) x, |x|,{x}} where , {x} denotes fractional part and [x] denotes greatest integer function , Let A, B , C are subsets of S. Suppose A : consists of odd functions (s) B : consists of discontinuous function (s) and C: consists of non-decreasing function(s) or increasing function (s). If f(x) in A nn C, g(x) in B nnC, h (x) in B" but not C and " l(x) in neither A nor B nor C . Then, answer the following. The range of f(h(x)) is

Let S denotes the set consisting of four functions and S = { [x], sin^(-1) x, |x|,{x}} where , {x} denotes fractional part and [x] denotes greatest integer function , Let A, B , C are subsets of S. Suppose A : consists of odd functions (s) B : consists of discontinuous function (s) and C: consists of non-decreasing function(s) or increasing function (s). If f(x) in A nn C, g(x) in B nnC, h (x) in B" but not C and " l(x) in neither A nor B nor C . Then, answer the following. The function f (x) is

Let f : R to R be a function defined as f(x) = {((b + 2)x-11c, x 2):} then f(x)is:

if f(x) and g(x) are continuous functions, fog is identity function, g'(b) = 5 and g(b) = a then f'(a) is

If both f(x) & g(x) are differentiable functions at x=x_0 then the function defiend as h(x) =Maximum {f(x), g(x)}

If f(x)a n dg(x) are continuous functions in [a , b] and are differentiable in (a , b) then prove that there exists at least one c in (a , b) for which. |f(a)f(b)g(a)g(b)|=(b-a)|f(a)f^(prime)(c)g(a)g^(prime)(c)|,w h e r ea

If f(x) is identity function, g(x) is absolute value function and h(x) is reciprocal function then (A) fogoh(x)=hogof(x) (B ) hog(x)=hogof(x) (C) gofofofohogof(x)=gohog(x) (D) hohohoh(x)=f(x)

if a function f(x) is (i) continuous on [a,b] (ii) derivable on (a,b) and (iii) f(x) gt 0 x in (a,b) then show that f(x) is strictly increasing on [a,b]