Suppose that `f` is differentiable for all `x` and that `f^(prime)(x)lt=2fora l lxdot` If `f(1)=2a n df(4)=8,t h e nf(2)` has the value equal to 3 (b) 4 (c) 6 (d) 8

If f(x)=(log)_x(lnx),t h e nf^(prime)(x) at x=e is equal to 1/e (b) e (c) 1 (d) zero

If f(x)=(log)_x(lnx),t h e nf^(prime)(x) at x=e is equal to 1/e (b) e (c) 1 (d) zero

Let f be differentiable for all x , If f(1)=-2a n df^(prime)(x)geq2 for all x in [1,6], then find the range of values of f(6)dot

If f(x) is a polynomial satisfying f(x)f(1/x)=f(x)+f(1/x)a n df(3)=28 ,t h e nf(4) is equal to 63 (b) 65 (c) 17 (d) none of these

Let f(x) be differentiable for real x such that f^(prime)(x)>0on(-oo,-4), f^(prime)(x) 0on(6,oo), If g(x)=f(10-2x), then the value of g^(prime)(2) is a. 1 b. 2 c. 0 d. 4

Let f:[1/2,1]->R (the set of all real numbers) be a positive, non-constant, and differentiable function such that f^(prime)(x)<2f(x)a n df(1/2)=1 . Then the value of int_(1/2)^1f(x)dx lies in the interval (a) (2e-1,2e) (b) (3-1,2e-1) (c) ((e-1)/2,e-1) (d) (0,(e-1)/2)

The function f: RvecR satisfies f(x^2)dotf^(x)=f^(prime)(x)dotf^(prime)(x^2) for all real xdot Given that f(1)=1 and f^(1)=8 , then the value of f^(prime)(1)+f^(1) is 2 b. 4 c. 6 d. 8

Suppose the function f(x)-f(2x) has the derivative 5 at x=1 and derivative 7a tx=2. The derivative of the function f(x)-f(4x)a tx=1 has the value equal to (a) 19 (b) 9 (c) 17 (d) 14

Suppose the function f(x) satisfies the relation f(x+y^3)=f(x)+f(y^3)dotAAx ,y in R and is differentiable for all xdot Statement 1: If f^(prime)(2)=a ,t h e nf^(prime)(-2)=a Statement 2: f(x) is an odd function.

If ("lim")_(tvecx)(e^tf(x)-e^xf(t))/((t-x)(f(x))^2)=2a n df(0)=1/2, then find the value of f^(prime)(0)dot a) 4 (b) 2 (c) 0 (d) 1