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.

If f(x) is a polynomial, show that between two roots of the polynomial equation f(x)=0, there lies at least one root of the polynomial f'(x)=0.

If x^2-(a-3)x+a =0 has at least one positive root,then

Between any two real roots of the equation e^xsinx-1=0 , the equation e^xcosx+1=0 has :

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 < b) . Show that there exists at least one root of equation f^(prime)(x)g^(prime)(x)+f(x)g(x)=0 on (a , b) .

If alpha and beta be two roots of the equation (x-a)(x-b)=c then show that a and b are the roots of (x-alpha)(x-beta) +c=0

Let f(x)=a_(5)x^(5)+a_(4)x^(4)+a_(3)x^(3)+a_(2)x^(2)+a_(1)x, "where" a_(i),s are real and f (x) = 0 has a positive root alpha_(0). Then f^(prime)(x)=0 has a positive root alpha_1 such that 0 < alpha_1 < alpha_0 f^(prime)(x)=0 has at least two real roots f^('')(x)=0 has at least one real root none of these

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

Show that the equation e^(x-1)+x-2=0 has no real root greater than 1.