Let `f:R->R` be a twice differential function such that `f(pi/4)=0,f((5pi)/4)=0 and f(3)=4`, then show that there exist a `cin(0,2pi)` such that `f^(primeprime)(c)+sinc-cosc<0`

Let f:Rto R be a twice continuously differentiable function such that f(0)=f(1)=f'(0)=0 . Then

Let f:[0,(pi)/(2)rarr[0,1] be a differentiable function such that f(0)=0,f((pi)/(2))=1 then

Let f:R rarr R be a twice differentiable function such that f(x+pi)=f(x) and f'(x)+f(x)>=0 for all x in R. show that f(x)>=0 for all x in R .

Let f: R->R be a twice differentiable function such that f(x+pi)=f(x) and f''(x)+f(x)geq0 for all x in Rdot Show that f(x)geq0 for all x in Rdot

If f(x) is continuous function in [0,2pi] and f(0)=f(2 pi ), then prove that there exists a point c in (0,pi) such that f(x)=f(x+pi).

If f(x) is continuous function in [0,2pi] and f(0)=f(2 pi ), then prove that there exists a point c in (0,pi) such that f(x)=f(x+pi).

If f(x) is continuous function in [0,2pi] and f(0)=f(2 pi ), then prove that there exists a point c in (0,pi) such that f(x)=f(x+pi).

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

Let f: \mathbb{R} rarr \mathbb{R} be a twice differentiable function such that f(x+pi)=f(x) and f''(x) + f(x) geq 0 for all x in \mathbb{R} . Show that f(x) geq 0 for all x in \mathbb{R} .