Let `f(x),xgeq0,` be a non-negative continuous function, and let `F(x)=int_0^xf(t)dt ,xgeq0,` if for some `c >0,f(x)lt=cF(x)` for all `xgeq0,` then show that `f(x)=0` for all `xgeq0.`

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

A continuous function f(x) satisfies the relation f(x)=e^x+int_0^1 e^xf(t)dt then f(1)=

Let f(x) be a non-positive continuous function and F(x)=int_(0)^(x)f(t)dt AA x ge0 and f(x) ge cF(x) where c lt 0 and let g:[0, infty) to R be a function such that (dg(x))/(dx) lt g(x) AA x gt 0 and g(0)=0 The number of solution(s) of the equation |x^(2)+x-6|=f(x)+g(x) is/are

Let f(x) be a non-positive continuous function and F(x)=int_(0)^(x)f(t)dt AA x ge0 and f(x) ge cF(x) where c lt 0 and let g:[0, infty) to R be a function such that (dg(x))/(dx) lt g(x) AA x gt 0 and g(0)=0 The total number of root(s) of the equation f(x)=g(x) is/ are

Let f(x) be a non-positive continuous function and F(x)=int_(0)^(x)f(t)dt AA x ge0 and f(x) ge cF(x) where c lt 0 and let g:[0, infty) to R be a function such that (dg(x))/(dx) lt g(x) AA x gt 0 and g(0)=0 The solution set of inequation g(x)(cos^(-1)x-sin^(-1)) le0

Let fa n dg be differentiable on R and suppose f(0)=g(0)a n df^(prime)(x)lt=g^(prime)(x) for all xgeq0. Then show that f(x)lt=g(x) for all xgeq0.

Let f: RvecR be a continuous function which satisfies f(x)= int_0^xf(t)dtdot Then the value of f(1n5) is______

Let f(x) be a differentiable function such that f(x)=x^2 +int_0^x e^-t f(x-t) dt then int_0^1 f(x) dx=

Let f(x) be a continuous function AAx in R , except at x=0, such that g(x)= int_x^a(f(t))/t dt , prove that int_0^af(x)dx=int_0^ag(x)dx

Let f:[0,1] rarr [0,1] be a continuous function. Then prove that f(x)=x for at least one 0lt=xlt=1.