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)=(x+3)/(5x^2+x-1) and g(x)=(2x+3x^2)/(20+2x-x^2) such that f(x) and g(x) are differentiable functions in their domains, then which of the following is/are true (a) 2f^(prime)(2)+g^(prime)(1)=0 (b) 2f^(prime)(2)-g^(prime)(1)=0 (c) f^(prime)(1)+2g^(prime)(2)=0 (d) f^(prime)(1)-2g^(prime)(2)=0

If f(x)=|logx|,xgt0 ,find f^(prime)(1/e)a n df^(prime)(e)

If f(x-y), f(x) f(y) and f(x+y) are in A.P. for all x , y ,and f(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^(prime)(x)=asinx+bcosx and f^(prime)(0)=4,f(0)=3,f(pi/2)=5 . Find f(x).

If f^(prime)(x)=asinx+bcosx and f^(prime)(0)=4,\ \ f(0)=3,\ \ f(pi/2)=5 , find f(x) .

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 g(x) be the inverse of an invertible function f(x) which is differentiable at x=c . Then g^(prime)(f(x)) equal. (a) f^(prime)(c) (b) 1/(f^(prime)(c)) (c) f(c) (d) none of these

If f(x)=m x+ca n df(0)=f^(prime)(0)=1. What is f(2)?

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

If f^(prime)(x)=x-1/(x^2) and f(1)=1/2 , find f(x) .