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

Let f(x)a n dg(x) be two functions which are defined and differentiable for all xgeqx_0dot If f(x_0)=g(x_0)a n df^(prime)(x)>g^(prime)(x) for all x > x_0, then prove that f(x)>g(x) for all x > x_0dot

If f(x)=x|x|, then prove that f^(prime)(x)=2|x|

If f(x)=|x|^(|sinx|), then find f^(prime)(-pi/4)

If f(x)=xsinx, then find f^(prime)(pi/2)

Solve x dy=(y+x(f(y/x))/(f^(prime)(y/x)))dx

Let (f(x+y)-f(x))/2=(f(y)-a)/2+x y for all real xa n dydot If f(x) is differentiable and f^(prime)(0) exists for all real permissible value of a and is equal to sqrt(5a-1-a^2)dot Then (a) f(x) is positive for all real x (b) f(x) is negative for all real x (c) f(x)=0 has real roots (d)Nothing can be said about the sign of f(x)

If intg(x)dx=g(x) , then evaluate intg(x){f(x)+f^(prime)(x)}dx

Suppose that f is differentiable for all x and that f^(prime)(x)lt=2 for all x If f(1)=2 and f(4)=8 then f(2) has the value equal to (a) 3 (b) 4 (c) 6 (d) 8

Let f(x+y)=f(x)dotf(y) for all xa n dydot Suppose f(5)=2a n df^(prime)(0)=3. Find f^(prime)(5)dot