Verify that `x sgnx=|x|`; `|x|sgnx=x`; `x(sgnx)(sgnx)=x`

Verify that x ^(3) + y ^(3) + z ^(3) - 3xyz =1/2 (x + y + z) [(x-y)^(2) + (y-z) ^(2) + (z-x) ^(2) ]

verify the existence of lim_(x to 1) f (x) , where f(x) ={{:((|x-1|)/(x-1) , "for" x ne 1),(0 , ", for " x=1):}

Verify the existence of lim_ (x to 1)f(x) , where f(x)={{:((|x-1|)/(x-1),"for" x ≠ 1),(0,"for" x=1):}

Verify the differential equation y = x sin x : xy' = y + x sqrt(x^(2) - y^(2))(x ne 0 and x gt y or x lt -y)

Verify the existence of lim_(x to 3) f(x) , where f(x)={:{(|x-3|/(x-3),for , xne 3),(0, for ,x =3):}

Verify that the function y=a cos x + b sin x is a solution of the differential equation cos x(dy)/(dx)+y sin x =b

Verify that 1,-1 and +3 are the zeroes of the cubic polynomial x^(2)-3x^(2)-x+3 and check the relationship between zeroes and the coefficients.

Verify Lagrange 's law of the mean for f(x) = x^(3) on [-2, 2].