If `g(x)=f(x)+f(1-x) and f prime prime (x)< 0; 0 <= x <= 1.` show that g(x) increases in `[0,1/2)` and decreases in `(1/2,1]`

Suppose that f(x) isa quadratic expresson positive for all real xdot If g(x)=f(x)+f^(prime)(x)+f^(prime prime)(x), then for any real x

Suppose that f(x) isa quadratic expresson positive for all real xdot If g(x)=f(x)+f^(prime)(x)+f^(prime prime)(x), then for any real x

Suppose that f(x) is a quadratic expresson positive for all real x If g(x)=f(x)+f^(prime)(x)+f^(prime prime)(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

Given H(x)=f(x).phi(x) and f prime (x).phi prime(x)=c then prove that (F prime prime prime)/F=(f prime prime prime)/f+(phi prime prime prime)/phi, where c in constant, when f(x),phi(x),F(x) are differentiable

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

A nonzero polynomial with real coefficient has the property that f(x)=f^(prime)(x)dot f^(prime prime)(x)dot If a is the leading coefficient of f(x), then the value of 1/(2a) is____

If f(x) attains a local minimum at x=c , then write the values of f^(prime)(c) and f^(prime)prime(c) .

If f(0)=f(1)=0 , f^(prime)(1)=f^(prime)(0)=2 and y=f(e^x)e^(f(x)) , write the value of (dy)/(dx) at x=0 .

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))