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)

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

Let g^(prime)(x)>0a n df^(prime)(x) g(f(x-1)) f(g(x+1))>f(g(x-1)) g(f(x+1))

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

Statement 1: For f(x)=sinx ,f^(prime)(pi)=f^(prime)(3pi)dot Statement 2: For f(x)=sinx ,f(pi)=f(3pi)dot

Let f be a continuous, differentiable, and bijective function. If the tangent to y=f(x)a tx=a is also the normal to y=f(x)a tx=b , then there exists at least one c in (a , b) such that (a) f^(prime)(c)=0 (b) f^(prime)(c)>0 (c) f^(prime)(c) (d) none of these

A function f: Rvec[1,oo) satisfies the equation f(x y)=f(x)f(y)-f(x)-f(y)+2. If differentiable on R-{0}a n df(2)=5,f^(prime)(x)=(f(x)-1)/xdotlambdat h e nlambda= 2^(prime)f(1) b. 3f^(prime)(1) c. 1/2f^(prime)(1) d. f^(prime)(1)

Let f(x+y)=f(x)dotf(y) for all xa n dydot Suppose f(5)=2a n df^(prime)(0)=3. Find f^(prime)(5)dot

Let f(x) be a non-constant twice differentiable function defined on (-oo,oo) such that f(x)=f(1-x)a n df^(prime)(1/4)=0. Then (a) f^(prime)(x) vanishes at least twice on [0,1] (b) f^(prime)(1/2)=0 (c) int_(-1/2)^(1/2)f(x+1/2)sinxdx=0 (d) int_(-1/2)^(1/2)f(t)e^(sinpit)dt=int_(1/2)^1f(1-t)e^(sinpit)dt