let `f(x)=2+cosx` for all real x Statement 1: For each real t, there exists a pointc in `[t,t+pi]` such that `f'(c)=0` Because statement 2: `f(t)=f(t+2pi)` for each real t

Let F(x) be an indefinite integral of sin^(2)x Statement-1: The function F(x) satisfies F(x+pi)=F(x) for all real x. because Statement-2: sin^(3)(x+pi)=sin^(2)x for all real x. A) Statement-1: True , statement-2 is true, Statement -2 is not a correct explanation for statement -1 c) Statement-1 is True, Statement -2 is False. D) Statement-1 is False, Statement-2 is True.

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)-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 exist some x in IR such that 2f(x) + 1 = 2x(1+x) Then

Let f(x) be differentiable on the interval (0,oo) such that f(1)=1 and lim_(t->x) (t^2f(x)-x^2f(t))/(t-x)=1 for each x>0 . Then f(x)=

Let F(x) be an indefinite integral of sin^(2)x Statement I The function F(x) satisfies F(x+pi)=F(x) for all real x. Because Statement II sin^(2)(x+pi)=sin^(2)x, for all real x.

If f(x) is an integrable function over every interval on the real line such that f(t+x)=f(x) for every x and real t, then int_(a)^(a+t) f(x)dx is equal to

If f(x) = ∫et^2(t - 2)(t - 3)dt for t ∈ [0, x] for x ∈ (0, ∞), then (A) f has a local maximum at x = 2 (B) f is decreasing on (2, 3) (C) there exists some c∈(0, ∞) such that f′′(c) = 0 (D) f has a local minimum at x = 3.

Let f be a continuous function on [a , b]dot If F(x)=(int_a^xf(t)dt-int_x^bf(t)dt)(2x-(a+b)), then prove that there exist some c in (a , b) such that int_a^cf(t)dt-int_c^bf(t)dt=f(c)(a+b-2c)dot

Given f(x) = 2/(x + 1) , what is(are) the real value(s) of t for which f(t) = t ?

"I f"f(x)={x^3-6x^2; ifxi sr a t ion a l-11 x+6;ifxi si r r a t ion a l"t h e n" f(x) is continuous for all real x f(x) is continuous at x=1,2,3. f(x) is discontinuous at infinite points f(x) discontinuous for all real xdot

Let f(x) be a continuous and differentiable function such that f(x)=int_0^xsin(t^2-t+x)dt Then prove that f^('')(x)+f(x)=cosx^2+2xsinx^2