Let `f:R rarr R` be a differentiable function such that its derivative `f` 'is continuous and `f(pi)=-6.` If `F:[0,1]rarr R` is defined by `F(x)=int_(0)^(x)f(t)dt` and if `int(f'(x)+F(x))cos xdx=2` ,then the value of `f(0)` is

Let R to R be a differentiable functions such that its derivative f' is continuous and f(pi) = - 6 . If F : [0,pi] to R is defined by F(x) int_(0)^(x) f(t) dt , and int_(0)^(x) f(t) dt and if int_(0)^(pi) (f'(x) +F(x))cos xdx = 2 then the value of f(0) is _________

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

if f:R rarr R is a differentiable function such that f'(x)>2f(x) for all x varepsilon R, and f(0)=1, then

f(x)=int_(0)^( pi)f(t)dt=x+int_(x)^(1)tf(t)dt, then the value of f(1) is (1)/(2)

If int_(0)^(x)f(t)dt=x+int_(x)^(1)f(t)dt ,then the value of f(1) is

If f:[0,pi]rarr R is continuous and int_(0)^( pi)f(x)sin xdx=int_(0)^( pi)f(x)cos xdx=0 then the number of roots of f(x) in (0,pi) is ...

Let f:R rarr R be a differential function satisfy f(x)=f(x-y)f(y)AA x,y in R and f'(0)=a,f'(2)=b then f'(-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

Let F(x)= F(x)=int_0^xtf(t)dt] and F(x^2)=x^4+x^5 then find the value of sum_(r=1)^12f(r^2)

Let f(x) be a continuous function defined from [0,2]rarr R and satisfying the equation int_(0)^(2)f(x)(x-f(x))dx=(2)/(3) then the value of 2f(1)