Commutation relations and the uncertainty principle

Consider two hermitian operators A and B which satisfy the following commutation relation:
[A, B] = AB-BA=iC, where C is also a hermitian operator in general. Let us introduce a new operator Q defined by: Q=A+ iλB, with λ being a real number, and consider the following scalar product:

Where is any normalized wave function?

(a)Show that Eq. (1) leads to the following result:
(b)Define the uncertainties as follows: With U≡A - and V≡B - < B>, respectively. Show that [U, V] = [A, B] = iC
(c)Thus, replacing A, B in (a) by U,V, we obtain: Regarding I(λ) as a quadratic function ofλ, show that I(λ) is minimum when
(d)Let A=X, B= px, show that the quantum condition [x, px]= i  leads to the uncertainty principle:

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