3. This question explores a general method for showing that a set of functions are linearly independent. We will only look at the case of three functions, but the idea and the arguments work for any number of functions. Given three functions f1, f2, and f3 6 C(]R), we dene their Wronskian to be f1 f2 f3 W(f1!f2af3)= fir f f1; 1 2 3 where f' and f\" mean the rst and second derivative of f, and the vertical bars mean to take the determinant. Note that the Wronskian (the determinant of the matrix above) is a flmction. What we want to prove is this: If W(f1, f2, f3) 7 0 (i.e., is not the zero flmction) then f1, f2, and f3 are linearly independent. (a) To get a feeling for the denition, compute W(a:, sinks), cos(a:)). Now let f1, f2, f3 6 (POUR) be any functions, and let's try and prove the statement above. (b) Explain why, if W(f1, f2, f3) 34$ 0, there must be an :50 E R such that W(f1, f2, f3)(;z:0) a 0. (0) Explain why, for this value of :30, f1($o) f2($o) f3($o) filIBO) Elma) films) {'(30) 3050) 51%) 7&0. (d) Now suppose that Cl f1 + szz + c3f3 = 0. Differentiate this relation twice, and plug 3:0 into each of the three relations to get three equations for c1, 02, and c3. (e) Explain why the only solution to the equations in (d) is c1 = 02 = 03 = 0, and hence that f1, f2, f3 are linearly independent