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.

Statement-1: The equation e^(x-1) +x-2=0 has only one real root. Statement-2 : Between any two root of an equation f(x)=0 there is a root of its derivative f'(x)=0

The equation sin x + x cos x = 0 has at least one root in

Between any two real roots of the equation e^(x)sinx-1=0 the equation e^(x)cosx+1=0 has

Let a and b be two distinct roots of a polynomial equation f(x) =0 Then there exist at least one root lying between a and b of the polynomial equation

Statement -1 : one root of the equation x^(2)+5x-7=0 lie in the interval (1,2). and Statement -2 : For a polynomial f(x),if f(p)f(q) lt 0, then there exists at least one real root of f(x) =0 in (p,q)

The equation sinx+x cos x=0 has atleast one root in the interval.

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

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: There is a value of k for which the equation x^(3) - 3x + k = 0 has a root between 0 and 1. Statement-2: Between any two real roots of a polynomial there is a root of its derivation.