Let `y=f(x)` be a parabola, having its axis parallel to the y-axis, which is touched by the line `y=x` at `x=1.` Then, (a)`2f(0)=1-f^(prime)(0)` (b) `f(0)+f^(prime)(0)+f^(0)=1` (c)`f^(prime)(1)=1` (d) `f^(prime)(0)=f^(prime)(1)`

If f(x-y),f(x)f(y),a n df(x+y) are in A.P. for all x , y ,a n df(0)!=0, then (a)f(4)=f(-4) (b)f(2)+f(-2)=0 (c)f^(prime)(4)+f^(prime)(-4)=0 (d)f^(prime)(2)=f^(prime)(-2)

If f: RvecR is a twice differentiable function such that f"(x)>0fora l lxR ,a n d f(1/2)=1/2,f(1)=1 then: f^(prime)(1)>1 (b) f^(prime)(1)lt=0 (c)1/2 ltf^(prime)(1)lt1 (d)="" 0ltf^(prime)(1)lt="1/2

Let g(x) be the inverse of an invertible function f(x), which is differentiable for all real xdot Then g^('')(f(x)) equals. (a) -(f^('')(x))/((f^'(x))^3) (b) (f^(prime)(x)f^('')(x)-(f^(prime)(x))^3)/(f^(prime)(x)) (c) (f^(prime)(x)f^('')(x)-(f^(prime)(x))^2)/((f^(prime)(x))^2) (d) none of these

Let the function f satisfies f(x)*f^(prime)(-x)=f(-x)*f^(prime)(x) foe all x and f(0)=3 The value of f(x).f^(prime)(-x)=f(-x).f^(prime)(x) for all x, is

If f(x)=|x-a|varphi(x), where varphi(x) is continuous function, then (a) f^(prime)(a^+)=varphi(a) (b) f^(prime)(a^-)=-varphi(a) (c) f^(prime)(a^+)=f^(prime)(a^-) (d) none of these

Let f(x+y)=f(x)f(y) for all x and y . Suppose f(5)=2 and f^(prime)(0)=3 , find f^(prime)(5) .

If f is a continuous function on [0,1], differentiable in (0, 1) such that f(1)=0, then there exists some c in (0,1) such that cf^(prime)(c)-f(c)=0 cf^(prime)(c)+cf(c)=0 f^(prime)(c)-cf(c)=0 cf^(prime)(c)+f(c)=0

Let y=f(x) be a function such that f'(x)=x^(3) and the line x+y=0 is tangent to the graph of f(x) then which of the following alternative(s) is/are correct? f(0)=-(3)/(4) (b) f(1)=1f(-1)=1 (d) f(3)=21

Consider the function f:(-oo,\ oo)->(-oo,oo) defined by f(x)=(x^2-a x+1)/(x^2+a x+1),\ 0

If f(x)<0,\ x in (a , b) then at the point C(c ,\ f(c)) on y=f(x) for which F(c) is a maximum, f^(prime)(c) is given by a. f^(prime)(c)=(f(b)-f(a))/(b-1) b. \ f^(prime)(c)=(f(b)-f(a))/(a-b) c. f^(prime)(c)=(2(f(b)-f(a)))/(b-a) d. f^(prime)(c)=0