For every twice differentiable function `f: R->[-2,\ 2]` with `(f(0))^2+(f^(prime)(0))^2=85` , which of the following statement(s) is (are) TRUE? There exist `r ,\ s in R` where `roo)f(x)=1` (d) There exists `alpha in (-4,\ 4)` such that `f(alpha)+f"(alpha)=0` and `f^(prime)(alpha)!=0`

For all twice differentiable functions f : R to R , with f(0) = f(1) = f'(0) = 0

Let f:(0, pi)->R be a twice differentiable function such that (lim)_(t->x)(f(x)sint-f(x)sinx)/(t-x)=sin^2x for all x in (0, pi) . If f(pi/6)=-pi/(12) , then which of the following statement(s) is (are) TRUE? f(pi/4)=pi/(4sqrt(2)) (b) f(x)<(x^4)/6-x^2 for all x in (0, pi) (c) There exists alpha in (0, pi) such that f^(prime)(alpha)=0 (d) f"(pi/2)+f(pi/2)=0

If f(0)=f(1)=f(2)=0 and function f (x) is twice differentiable in (0, 2) and continuous in [0, 2]. Then, which of the following is(are) true ?

Let f:R rarr R be a twice differential function such that f((pi)/(4))=0,f((5 pi)/(4))=0 and f(3)=4 then show that there exist a c in(0,2 pi) such that f'(c)+sin c-cos c<0

If f:R->R is a twice differentiable function such that f''(x) > 0 for all x in R, and f(1/2)=1/2 and f(1)=1, then

Let f be a twice differentiable function on [0,2] such that f(0)=0,f(1)=2,f(2)=4, then prove that f^(prime)(alpha)=2\ for\ som e\ alpha\ in (0,1) f^(prime)(beta)=2\ for\ som e\ beta\ in (1,2) f' '(gamma)=0\ for\ som e\ gamma\ in (0,2)

Let f:[0,4]rarr R be a differentiable function then for some alpha,beta in (0,2)int_(0)^(4)f(t)dt

If for some differentiable function f(alpha)=0a n df^(prime)(alpha)=0, Statement 1: Then sign of f(x) does not change in the neighbourhood of x=alpha Statement 2: alpha is repeated root of f(x)=0