Let `f(x) = (1-x)^2 sin^2x + x^2` for all `x in IR,` and let `g(x)=int_1^x((2(t-1))/(t-1)- lnt) f(t)` dt for all `x, in (1,oo).` Consider the statements: P: There exists some `x in IR` such that `f(x) +2x=2(1+x^2)` Q: There exists some `x in IR` such that `2f(x) +1 =2x(1+x)` Then

Let f(x)=(1-x)^(2)sin^(2)x+x^(2) for all x in IR, and let g(x)=int_(1)^(x)((2(t-1))/(t-1)-ln t)f(t) dt for all x,in(1,oo). Which of the following is true?

Let f:[0,1]toR (the set of all real numbers ) be a function. Suppose the function f is twice differentiable,f(0)=f(1)=0 and satisfies f''(x)-2f'(x)+f(x)gee^(2),x in [0,1] Consider the statements. I. There exists some x in R such that, f(x)+2x=2(1+x^(2)) (II) There exists some x in R such that, 2f(x)+1=2x(1+x)

Let f(x) = (x+2)^3 , for -3 < x <= -1 , x^(2/3) , for -1 < x < 2 and g(x) = int_-3^x f(t) dt for all x in (-3, 2) Find the nos of extremum points of g'(x)

Let f (x)= int _(x^(2))^(x ^(3))(dt)/(ln t) for x gt 1 and g (x) = int _(1) ^(x) (2t ^(2) -lnt ) f(t) dt(x gt 1), then:

Let f(x)=int_(x^(2))^(x^(3)) for x>1 and g(x)=int_(1)^(x)(2t^(2)-Int)f(t)dt(x<1) then

Let f(x)=int_(1)^(x)(3^(t))/(1+t^(2))dt , where xgt0 , Then

Let f(x)=int_(0)^(x)(t-1)(t-2)^(2)"dt. If "f(x) gek for all x and for some k, then the set of exhaustive value of k is