Show that between any two roots of `e^(-x)-cosx=0,` there exists at least one root of `sinx-e^(-x)=0`

Which of the following is/are correct? (A) Between any two roots of e^xcosx=1, there exists at least one root of tanx=1. (B) Between any two roots of e^xsinx=1, there exists at least one root of tanx=-1. (C) Between any two roots of e^xcosx=1, there exists at least one root of e^xsinx=1. (D) Between any two roots of e^xsinx=1, there exists at least one root of e^xcosx=1.

Let f(x)a n dg(x) be differentiable functions such that f^(prime)(x)g(x)!=f(x)g^(prime)(x) for any real xdot Show that between any two real solution of f(x)=0, there is at least one real solution of g(x)=0.

Let f(x) be differentiable function and g(x) be twice differentiable function. Zeros of f(x),g^(prime)(x) be a , b , respectively, (a

Let f(x)=a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x , where a_i ' s are real and f(x)=0 has a positive root alpha_0dot Then f^(prime)(x)=0 has a positive root alpha_1 such that 0

Statement 1: If 27 a+9b+3c+d=0, then the equation f(x)=4a x^3+3b x^2+2c x+d=0 has at least one real root lying between (0,3)dot Statement 2: If f(x) is continuous in [a,b], derivable in (a , b) such that f(a)=f(b), then there exists at least one point c in (a , b) such that f^(prime)(c)=0.

Statement 1: If g(x) is a differentiable function, g(2)!=0,g(-2)!=0, and Rolles theorem is not applicable to f(x)=(x^2-4)/(g(x))in[-2,2],t h e ng(x) has at least one root in (-2,2)dot Statement 2: If f(a)=f(b),t h e ng(x) has at least one root in (-2,2)dot Statement 2: If f(a)=f(b), then Rolles theorem is applicable for x in (a , b)dot

Show that the equation e^(sinx)-e^(-sinx)-4=0 has no real solution.

Integrate the functions e^(x)(sinx+cosx)dx

If 3(a+2c)=4(b+3d), then the equation a x^3+b x^2+c x+d=0 will have (a)no real solution (b)at least one real root in (-1,0) (c)at least one real root in (0,1) (d)none of these