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

Show that between any two roots of e^(x) cos x=1 , there exists at least one root of e^(x) sin x-1=0

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

Let f(x)=0 be a polynomial equation with real coefficients. Then between any two distinct real roots of f(x)=0 , there exists at least one real root of the equation f'(x)=0 . This result is a consequence of the celebrated Rolle's theorem applied to polynomials. Much information can be extracted about the roots of f(x)=0 from the roots of f'(x)=0 . If the three roots of x^(3)-12x+k=0 lie in intervals (-4,-3), (0,1) and (2,3) ,then the exhaustive range of values of k is

Let f(x)=0 be a polynomial equation with real coefficients. Then between any two distinct real roots of f(x)=0 , there exists at least one real root of the equation f'(x)=0 . This result is a consequence of the celebrated Rolle's theorem applied to polynomials. Much information can be extracted about the roots of f(x)=0 from the roots of f'(x)=0 . The range of values of k for which the equation x^(4)+4x^(3)-8x^(2)+k=0 has four real and unequal roots is

Let f(x)=0 be a polynomial equation with real coefficients. Then between any two distinct real roots of f(x)=0 ,there exists at least one real root of the equation f'(x)=0 . This result is a consequence of the celebrated Rolle's theorem applied to polynomials. Much information can be extracted about the roots of f(x)=0 from the roots of f'(x)=0 . Q.The exhaustive range of values of k for which the equation x^(4)-14x^(2)+24x-k=0 has four unequal real roots is

Let f(x)=0 be a polynomial equation with real coefficients. Then between any two distinct real roots of f(x)=0 , there exists at least one real root of the equation f'(x)=0 . This result is a consequence of the celebrated Rolle's theorem applied to polynomials. Much information can be extracted about the roots of f(x)=0 from the roots of f'(x)=0 . The range of values of k for which the equation x^(4)-14x^(2)+24x-k=90 has four unequal real roots is

In between any two real roots of an e^(x)sin x=1 there exists how many roots satisfying equation e^(x)cos x=-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