Question

If 3.80 fig: [a,b] → R 2 Alonspiciens differentiable functions and we suppose Fca) = f(b) =. The wronskien of these a functions is the function TW Cf. g): [a, b] R defined by wCfg) () = det (FX) 906) -F68) g'(x)=9(x)}f'(X) (f'(x) g(x)) If W (f, g) (x) #0 for all x E [a,b], show that it exist a c E Ca,b) such that g (c) = 0.

Answer 1

is differentimble, hence contincions Given, of g [a, b] R are differentiable functions with f(a) = f(b)= 0. Wronskian of these two functions & g are W(fy): [a, b] R W (f, g) (n) = det off) g) flug'n-growife f(x) glin) defined by If possible, let g(x) = 0 xxf(a,b) Now if g (a) = 0, Since, Consider the Can We ffa) Vre[a,b] w (f, 9) (a)= f(a) g'(a)- g(a) f(a) which contradicts the fact W (f, g) (2) 40 vnf[a,b] g Ca 40. Similarely g (b) +0, (2) #0 HX € [a,b], so function (4): [4,63 R , )(2) (2) is els continuous on [a,b], differentiable [a, b] (since of g are Ja (5) Ca) = (%)-) = 0 By Rolled theorem, Je t'))-greute (f) (CA)= Bo, but [g (CA)] ² - W (f») ce? [g (c)] (33%CC")=0 O =) N (f, y) (*)= 0, where e* E (, ») sol y * е (a,b) such that I cany which leads to to a contradiction. Hence, I c Ella,b) such that gle)