Let f(x) be polynomial function of degree 2 such that `f(x)gt0` for all `x in R.` If `g(x)=f(x)+f'(x)+f''(x)` for all x, then

If f(x) is a polynomial function f:R→R such that f(2x)=f (x) f ′′(x) Then f(x) is

Let f be a function such that f(x+y)=f(x)+f(y) for all xa n dya n df(x)=(2x^2+3x)g(x) for all x , where g(x) is continuous and g(0)=3. Then find f^(prime)(x)dot

If f(x) g(x) and h(x) are three polynomials of degree 2 and Delta = |( f(x), g(x), h(x)), (f'(x), g'(x), h'(x)), (f''(x), g''(x), h''(x))| then Delta(x) is a polynomial of degree (dashes denote the differentiation).

Let f(x) be a polynomial of degree four having extreme values at x =1 and x=2. If underset(xrarr0)lim[1+(f(x))/x^2] =3, then f(2) is equal to

Let f(x) be a polynomial of degree 5 such that f(|x|)=0 has 8 real distinct , Then number of real roots of f(x)=0 is ________.

Let F:RtoR be a thrice differntiable function. Suppose that F(1)=0,F(3)=-4 and F'(x)lt0 for all x epsilon(1,3) . Let f(x)=xF(x) for all x inR . Then the correct statement(s) is (are)

Let f be a differentiable function satisfying [f(x)]^(n)=f(nx)" for all "x inR. Then, f'(x)f(nx) a. f(x) b. 0 c. f(x)f'(nx) d. none of these

Let f be a function defined on [a, b] such that f'(x)>0 for all x in (a,b) . Then prove that f is an increasing function on (a, b) .

Let f(x) be a continuous function such that f(0) = 1 and f(x)-f (x/7) = x/7 AA x in R , then f(42) is

Let f:(0,oo)rarrR be a differentiable function such that f'(x)=2-(f(x))/(x) for all x in (0,oo) and f(1) ne 1. Then