Let `y=f(x)` be drawn with `f(0) =2` and for each real number `a` the line tangent to `y = f(x)` at `(a,f(a))` has x-intercept ` (a-2)`. If `f(x)` is of the form of `k e^(px)` then`k/p` has the value equal to

If f(x) = cos(log x) then f(x)f(y)-1/2[f(x/y)+f(xy)] has the value

If f(x + y) = f(x) +f(y), then the value of f(0)

Let f(x) = {-x^2 ,for x<0; x^2+8 ,for xgeq0 Find x intercept of the common tangent to f(x) .

Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1)=1. If the y- intercept of the tangent at any point P(x , y) on the curve y=f(x) is equal to the cube of the abscissa of P , then the value of f(-3) is equal to________

If f(x) = cos (log _(e) x), then f (x) f(y) -1/2 [f(xy) +f((x)/(y))] has the value-

If f(x) = (a^x + a^-x)/2, (a gt 2) then the value of f(x + y)+f(x - y) is equal to

Let f(x) be periodic and k be a positive real number such that f(x+k)+f(x)=0fora l lx in Rdot Prove that f(x) is periodic with period 2kdot

Let f(x+y) = f(x) . f(y) for all x,y where f(0) ne 0. If f(5) =2 and f'(0) =3 , then f'(5) is equal to

If f(x)f(y) = f(x) + f(y) + f(xy) -2 for all real values of x and y and f(2) = 5, find f(4) and f(1/4) .

Let f be a function such that f(x + y) = f(x) + f(y) forall x , y in R , if f(1) = k, then show that f(n) is equal to nk.