Let `f` be a twice differentiable function such that `f(a)=f(b)=0` and `f(c)>0` for `a < c < bdot` Prove that there exists at least one value `lambda` between `aa n db` fow which `f^(lambda)<0.`

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 rarr R be a differentiable function such that f(a)=0=f(b) and f'(a)f'(b)>0 for some a

Let f be a twice differentiable function such that f''(x)=-f(x)" and "f'(x)=g(x)."If "h'(x)=[f(x)]^(2)+[g(x)]^(2), h(a)=8" and "h(0)=2," then "h(2)=

If f(x) is a twice differentiable function such that f(0)=f(1)=f(2)=0 . Then

Let f be a twice differentiable function such that f''(x)=1-f(x) where f'(0)=f(0)=1 The value of int_(0)^((pi)/(2))(f(x)-1)/(cos^(2)x+1) is equal to

Let f be a differentiable function such that f(0)=e^(2) and f'(x)=2f(x) for all x in R If h(x)=f(f(x)) ,then h'(0) is equal to

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

Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f '(x) ne 0 for all x in R . If |[f(x)" "f'(x)], [f'(x)" "f''(x)]|= 0 , for all x in R , then the value of f(1) lies in the interval:

Let f be arry continuously differentiable function on [a,b] and twice differentiable on (a.b) such that f(a)=f(a)=0 and f(b)=0. Then

If f(x) is a twice differentiable function such that f(a)=0,f(b)=2,f(c)=-1,f(d)=2,f(e)=0 where a