If `F(x)=f(x)dotg(x)` and `f^(prime)(x)dotg^(prime)(x)=c ,` prove that `(F^(primeprime))/F=f^(primeprime)/f+g^(primeprime)/g+(2c)/(fg)&F^(primeprimeprime)/F=f^(primeprimeprime)/f+g^(primeprimeprime)/g`

Let g(x)=2f(x/2)+f(1-x) and f^(primeprime)(x)<0 in 0<=x<=1 then g(x)

Let f(0)=f'(0)=0 and f^(primeprime)(x)=sec^4x+4 then find f(x)

If f(x),g(x) and h(x) are three polynomials of degree 2, then prove that phi(x) = |[f(x),g(x),h(x)],[f ^ (prime)(x),g^(prime)(x),h^(prime)(x)],[f^(primeprime)(x),g^(primeprime)(x),h^(primeprime)(x)]| is a constant polynomial.

A function f: R->R satisfies sinxcosy(f(2x+2y)-f(2x-2y)=cosxsiny(f(2x+2y)+f(2x-2y))dot If f^(prime)(0)=1/2,t h e n (a) f^(primeprime)(x)=f(x)=0 (b) 4f^(primeprime)(x)+f(x)=0 (c) f^(primeprime)(x)+f(x)=0 (d) 4f^(primeprime)(x)-f(x)=0

If f(x)=|x-2| a n d g(x)=f[f(x)],t h e n g^(prime)(x)= ______ for x>20

Suppose f and g are functions having second derivative f'' and g' ' everywhere. If f(x)dotg(x)=1 for all x and f^(prime) and g' are never zero, then (f^('')(x))/(f^(prime)(x))-(g^('')(x))/(g^(prime)(x)) is equal (a) (-2f^(prime)(x))/f (b) (2g^(prime)(x))/(g(x)) (c) (-f^(prime)(x))/(f(x)) (d) (2f^(prime)(x))/(f(x))

If x=f(t)cost-f^(prime)(t)sint and y=f(t)sint+f^(prime)(t)cost , then ((dx)/(dt))^2+((dy)/(dt))^2= (a) f(t)-f^(primeprime)(t) (b) {f(t)-f^(primeprime)(t)}^2 (c) {f(t)+f^(primeprime)(t)}^2 (d) none of these

If f(x)=log_(e)x and g(x)=e^(x) , then prove that : f(g(x)}=g{f(x)}

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