Q4. The CauchySchwcrz Inequality says that if a = (cl....,on} and b 2 (b1... . ,bn) are two vectors in R\". then la'bl S llallllbll- This inequality has many practical applications. In this exercise you will prove it using multivariahle calculus! (a) Show that the inequality is trivially true if a or b is the zero vector. Thus in what follows we may assume that a and b are non-zero vectors. (b) Suppose you know that. the inequality is true for all unit vectors '3 E R\". Show that you can deduce from this that the inequality must be true for all vectors in E R\". [Hint: Consider the normalization b = ll-l (c) Find the maximum and minimum values of the function f(:2:1,...,:rn} = Z; air; subject to the constraint lell 2 c where c E R90 is a xed positive real number. [Hint: The Lagrange multipliers algorithm applies in the same way to a function of n variables subject to one constraint as it does to functions of 2 and 3 variables. You may use. without proof, the fact. that the set S = {x E R\": \"x\" = c} is closed and bounded and has no \"edge points."'] (d) Using your ndings in parts {a} and (b). give a proof of the CauchySchwarz Inequality