Let `f'(x)=(192x^3)/(2+sin^4pix)` for all `x in R` with f(1/2)=0.If `mleint_(1//2)^1f(x)dxleM`, then the possible values of m and M are

Let f'(x)=(192x^(3))/(2+sin^(4)pix) for all x in R with f((1)/(2))=0 . If m le int_((1)/(2))^(1)f(x)dx leM , then the possible values of m and M are-

Let f prime(x)=(192x^3)/(2+sin^4 pix) for all x in RR with f(1/2)=0. If mlt=int_(1/2)^1f(x)dxlt=M then for x in [(1/2),1] the possible values of m and M are

If x^2f(x)+f(1/x)=2 for all x except at x=0 then int_(1/3)^3 f(x) dx=

Find all the possible values of f(x) =(1-x^2)/(x^2+3)

Find all the possible values of f(x) =(1-x^2)/(x^2+3)

Let F:RrarrR be a thrice differentiable function. Suppose that F(1)=0, F(3)=-4 and F'(x)lt0 for all x in (1//2,3) . Let f(x)=xF(x) for all x in R . If int_1^3x^2F'(x)dx=-12 and int_1^3x^3F''(x)dx=40 , then the correct expression(s) is (are)

If f (2x+3) =sin x+2^(x), then the value of f (4m-2n+3) is-

Let f(x)=(1-x)^(2) sin^(2) x+x^(2) for all x in RR , and let g(x)=int_(1)^(x) ((2(t-1))/(t+1)-log t)f(t) dt for all x in (1, oo) . Consider the statements : P : there exists some x in RR such that f(x)+2x=2(1+x^(2)) Q : There exists some x in RR such that 2f(x)+1=2x(1+x) then-

Let f(x) be a differentiable function in [2,7]. If f(2) = 3 and f'(x)le5 all x in (2,7) then the maximum possible value of f(x) at x=7 is

Let d/(dx) (F(x))= e^(sinx)/x, x>0 . If int_1^4 (2e^sin(x^2))/x dx = F(k)-F(1) , then possible value of k is: