If `f, phi, psi` are continuous in [a, b] and derivable in ]a, b[ then show that there is a value of c lying between a & b such that, `|(f(a),f(b),f'(c)),(phi(a),phi(b),phi'(c)),(psi(a),psi(b),psi'(c))|=0`

int_(a)^(b)f(x)dx=phi(b)-phi(a)

If the functions f(x) and phi(x) are continuous in [a,b] and differentiable in (a,b), then the value of c'c' 'for the pair of functions f(x)=e^(x),phi(x)=e^(-x) is

If the function f(x) and phi (x) are continuous in [a,b] and differentiable in (a,b) , then the value of 'c' for the pair of functions f(x) = e^(x), phi (x) = e^(-x) is

If the functions f(x) and phi(x) are continuous in [a,b] and differentiable in (a,b), then the value of 'c' for the pair of functions f(x)=sqrt(x),phi(x)=(1)/(sqrt(x)) is

if the functions f(x) and phi (x ) are continuous in [a,b] and differentiable in (a,b) , then the value of 'c' for the pair of function f (x) = sqrt(x ) , phi (x) = ( 1)/( sqrt(x)) is

If A nn B^(c)= phi , show that A sub B .

If phi(x)=2^(mx+1) , show that phi(a). phi(b).phi(c)=4.phi(a+b+c).

Let f(x) be a function such that its derovative f'(x) is continuous in [a, b] and differentiable in (a, b). Consider a function phi(x)=f(b)-f(x)-(b-x)f'(x)-(b-x)^(2) A. If Rolle's theorem is applicable to phi(x) on, [a,b], answer following questions. If there exists some unmber c(a lt c lt b) such that phi'(c)=0 and f(b)=f(a)+(b-a)f'(a)+lambda(b-a)^(2)f''(c) , then lambda is