Question

***** Please just answer the Problem 2. The stuff above is just the information one might need. ******

"fodal 5]*fda)}([^ g?da). SI = $*f*da)ă, vfeR(a). Define Prove the following triangle inequality If – 9|| = || f – h|| + || h – 9||, Vf,g, h E R(a). Problem 2 [10] Prove that, suppose a(x) = x, and f,g are continuous on [0, 1], if the equality holds in (1), then there exist a, b e R, a² +62 € 0, such that a f () + bg(x) = 0, Vx € [0, 1].

Answer 1

Given that 106-10-09 (424) s dx = da comes a equeno be 2 Problem - 2: Griven Chat equality 1 0 49 4+2 | ****52-22) holds 1549+= 56°42)" (192) ( 19 ) C ode) {ret, A = 't de: B = fgr de ; C = ffg da/ consider P o = ((f - agede for some XER. rde - 2xlf g da tx A -2XC + B -BX - 20x ta. p=0 iff BX-2c&ta 20 2c + √40²-4AB 2B C & VEZ AB B . ICEAB

then В becomes Zero - so, if we choose - - - p=/f-iggde becomes a reso Tron, 4(x) = (for % gla) is a contineras function and s'h(x) = 0 non-negative so, k(x) = 0 (fa) - / gla) = 0 = f(a)- / 9 (2) = 0 =) B f(n)-C gato & Mon, choose a = B = f gr de and 6=-( = - (ig de then, from we get af(x) + 6 (x) = o.