If f(x) is a quadratic expression such that `f(x)gt 0 AA x in R`, and if `g(x)=f(x)+f'(x)+f''(x)`, then prove that `g(x)gt 0 AA x in R`.

Let f(x)=ax^2+bx+c,a,b,cepsilon R a !=0 such that f(x)gt0AAxepsilon R also let g(x)=f(x)+f\'(x)+f\'\'(x) . Then (A) g(x)lt0AAxepsilon R (B) g(x)gt0AAxepsilon R (C) g(x)=0 has real roots (D) g(x)=0 has non real complex roots

Let f''(x) gt 0 AA x in R and let g(x)=f(x)+f(2-x) then interval of x for which g(x) is increasing is

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

lf f'(x) > 0,f"(x)>0AA x in (0,1) and f(0)=0,f(1)=1 ,then prove that f(x)f^-1(x) lt x^2AA x in (0,1)

Let f(x) be a quadratic expression which is positive for all real x and g(x)=f(x)+f'(x)+f''(x) .A quadratic expression f(x) has same sign as that coefficient of x^2 for all real x if and only if the roots of the corresponding equation f(x)=0 are imaginary.Which of the following holds true? (A) g(0)g(1)lt0 (B) g(0)g(-1)lt0 (C) g(0)f(1)f(2)gt0 (D) f(0)f(1)f(2)lt0

Let g'(x)gt 0 and f'(x) lt 0 AA x in R , then

Let f(x) be a function satisfying f(x) + f(x+2) = 10 AA x in R , then

If f(x) is a differentiable function satisfying |f'(x)|le4AA x in [0, 4] and f(0)=0 , then

If g (x) = x ^(2) -1 and gof (x) = x ^(2) + 4x + 3, then f ((1)/(2)) is equal (f(x) gt 0 AA x in R)

If f(x) is a real valued function such that f(x + 6) - f(x + 3) + f(x) = 0, AA x in R , then period of f(x) is