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

Statement 1: If both functions f(t)a n dg(t) are continuous on the closed interval [1,b], differentiable on the open interval (a,b) and g^(prime)(t) is not zero on that open interval, then there exists some c in (a , b) such that (f^(prime)(c))/(g^(prime)(c))=(f(b)-f(a))/(g(b)-g(a)) Statement 2: If f(t)a n dg(t) are continuou and differentiable in [a, b], then there exists some c in (a,b) such that f^(prime)(c)=(f(b)-f(a))/(b-a)a n dg^(prime)(c)(g(b)-g(a))/(b-a) from Lagranes mean value theorem.

A function f is defined by f(x)=int_0^pi costcos(x-t)dt, 0lexle2pi . Which of the following hold(s) good? (A) f(x) is continuous but not differentiable in (0,2pi) (B) There exists at least one c in (0,2pi) such that f\'(c)=0 (C) Maximum value of f is pi/2 (D) Minimum value of f is -pi/2

Let f(x) = (1 - x)2sin2x + x2 for all x ∈ R, and let g(x) = ∫(2(t - 1)/(t + 1) - ln t)f(t)dt for t ∈ [1, x] for all x ∈ (1, ∞). Consider the statements: P: There exists some x ∈ R, such that f(x) + 2x = 2(1 + x2) Q: There exists some x ∈ R, such that 2f(x) + 1 = 2x(1 + x) (A) both P and Q are true (B) P is true and Q is false (C) P is false and Q is true (D) both P and Q are false.

Let f:RtoR be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt . f(x) increases for

Statement 1: Let f(x)=sin(cos x)in[0,pi/2]dot Then f(x) is decreasing in [0,pi/2] Statement 2: cosx is a decreasing function AAx in [0,pi/2]

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

Let (f(x+y)-f(x))/2=(f(y)-a)/2+x y for all real x and ydot If f(x) is differentiable and f^(prime)(0) exists for all real permissible value of a and is equal to sqrt(5a-1-a^2)dot Then a) f(x) is positive for all real x b) f(x) is negative for all real x c) f(x)=0 has real roots d) Nothing can be said about the sign of f(x)

Let f:RtoR be a differentiable function such that f(x)=x^(2)+int_(0)^(x)e^(-t)f(x-t)dt . y=f(x) 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:R -(0,oo) be a real valued function satisfying int_0^x tf(x-t) dt =e^(2x)-1 then f(x) is