Let `f(x),xgeq0,` be a non-negative continuous function. If `f^(prime)(x)cosxlt=f(x)sinxAAxgeq0,` then find `f((5pi)/3)`

Let f(x), xge0 , be a non-negative continous function,and let F(x) = int_(0)^(x)f(t)dt, xge0 , if for some cgt0,f(x)lec F(x) for all xge0 , then show that f(x)=0 for all xge0 .

Let f(x) be a non-negative continuous function such that the area bounded by the curve y=f(x), the x-axis, and the ordinates x=(pi)/(4) and x=betagt(pi)/(4)" is "beta sin beta +(pi)/(4)cos beta +sqrt(2)beta. Then f'((pi)/(2)) is

If f(x)=|x|^(|sinx|), then find f^(prime)(-pi/4)

Let f(x) be non- negative continuous function such that the area bounded by the curve y=f(x) , x- axis and the ordinates x= (pi)/(4) and x = beta ( beta gt (pi)/(4)) is beta sin beta + (pi)/(4) cos beta + sqrt(2) beta . Then the value of f((pi)/(2)) is-

If a continuous function f on [0, a] satisfies f(x)f(a-x)=1, a >0, then find the value of int_0^a(dx)/(1+f(x))

let f(x) be a continuous function such that f(a-x)+f(x)=0 for all x in[0,a] then int_0^a dx/(1+e^f(x) is equal to

Let f(x+y)=f(x)+f(y)+2x y-1 for all real x and y and f(x) be a differentiable function. If f^(prime)(0)=cosalpha, the prove that f(x)>0AAx in Rdot

Let f(x) be a continuous and periodic function such that f(x)=f(x+T) for all xepsilonR,Tgt0 .If int_(-2T)^(a+5T)f(x)dx=19(agt0) and int_(0)^(T)f(x)dx=2 , then find the value of int_(0)^(a)f(x)dx .

Let f(x0 be a non-constant thrice differentiable function defined on (-oo,oo) such that f(x)=f(6-x)a n df^(prime)(0)=0=f^(prime)(x)^2=f(5)dot If n is the minimum number of roots of (f^(prime)(x)^2+f^(prime)(x)f^(x)=0 in the interval [0,6], then the value of n/2 is___

Let f(x) be a derivable function f'(x)gtf(x)" and " f(0)=0 . Then