hove directly that the functions f1 (x) =1. fa(x) = x, and f2(x) = x2 are linearly independent on the whole real line. n me n functions f1, f2,..., fn are said to be linearly independent on the interval I provided that the identity Cif1 + C2f2 + .. + Cnfn =0 holds on I only when (1) - W art for constants C1, C2, .., Cn• Given, f1 (x) = 1, f2(x) = x, and fa(x) = x², the identity c, f, + c,f, + ... as, + Cnfn =0 can bewritten (0 CL= (-0. The current equation involves thee unknowns, c1. cɔ. and ca. Two more equations are needed in order to determine the Values of all three unknowns. How can these two equations be determined? Select the correct choice below and fill in the answer boxes to complete your choice. %3(1.0.JW nsilanoW Insbneqabni yhoenil ons r W aA O A. Determine the first and second derivatives of the current equation, with respect to x. The first derivative is (x)s (d.p.w nablandw eT B nebnoqab hsonil en enll isat s 0.0 gebou 3(d.p .W nelaniow erfT O The second derviative is = 0. O B. Find both roots of the current equation. Vlieofinsbi zi W A W erT O The smaller root is x = nobnoqabni yheend ons (0r 3 (rt a w nsixenow erfT a0 3nebreceu yheonil ss (X)r The larger root is x = orlt no 0 ylleoitnebi 2i W EA These three equations form a system of three unknowns. Because the solution to this system is c, = C2 = _ bma, and c3 = , the functions f1 (x) = 1, f2(x)= x, and f3(x) = x2 are linearly independent. (1) ○ C1#C2チ そCn#0, C1 = C2 = ... = Cn = 0, -Sh =0 M0)=rA,(0)=4 A (0)= 0 %3D 0s bnH vwoled nevig ou enohutoz tnebneqabm ylsenil sedt bna notsupe 1sen ebeugeu 20 oin uin nevig er paivteina nouloe

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