Let `f(x) = lambda + mu|x|+nu|x|^2`, where `lambda,mu, nu in R`, then `f'(0)` exists if

Verify that the function y = lambda x + (mu)/(x) where lambda, mu are arbitary constants is a solution of the differential equation x^(2)y'' + xy' - y = 0 .

Verify that the function y = 2lambda x + (mu)/(x) where lambda, mu are arbitary constants is a solution of the differential equation x^(2)y'' + xy' - y = 0 .

If f(x)={:{([x]+[-x] , xne 0), ( lambda , x =0):} where [.] denotes the greatest function , then f(x) is continuous at x =0 , " for" lambda

If f(x)={:{([x]+[-x] , xne 0), ( lambda , x =0):} where [.] denotes the greatest function , then f(x) is continuous at x =0 , " for" lambda

Let f: R to R be defined as f(x)={{:(x^(5) sin""((1)/(x))+5x^(2)",",x lt0),(" 0,", x=0),(x^(5) cos""((1)/(x))+ lambdax^(2)",",x gt 0):} The value of lambda for which f''(0) exists , is ____________

Let f(x) = [{:(5^(1//x),x lt 0),(lambda[x],x le 0","lambda in R):} , then at x = 0 :

If int 2/(2-x)^2 ((2-x)/(2+x))^(1//3)\ dx = lambda ((2+x)/(2-x))^mu + c where lambda and mu are rational number in its simplest form then (lambda+1/mu) is equal to

For a,b,in R and a!=0, the equation ax^(2)+bx+c=0 and x^(2)+2x+3=0 have atleast one common root if (a)/(lambda)=(b)/(mu)=(c)/(Psi) where lambda,mu and Psi are positive integers.The least value of (lambda+mu+Psi) is

Let f(x)={(5^(1//x)"," , x lt 0),(lambda[x]",",x ge0):} and lambda in R , then at x = 0