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)

Suppose that f(x) is a quadratic expresson positive for all real xdot If g(x)=f(x)+f^(prime)(x)+f^(primeprime)(x), then for any real x (where f^(prime)(x) and f^(primeprime)(x) represent 1st and 2nd derivative, respectively). (a) g(x) 0 (c). g(x)=0 (d). g(x)geq0

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

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

Let g(x)=2f(x/2)+f(x) and f''(x)lt0 " in "0lexle1 , then g(x)

Let g(x)=2f(x/2)+f(2-x) and f''(x)<0 , AA x in (0,2) .Then g(x) increasing in

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

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

Let f(x)=int_0^x(dt)/(sqrt(1+t^3))a n dg(x) be the inverse of f(x) . Then the value of 4(g^(primeprime)(x))/((g(x))^2)i s____

Let f(x+y)=f(x)f(y) and f(x) = 1 + (sin2x) g(x) where g(x) is continuous . Then f^(1) (x) equals