Lecture Notes Course à MA 190 Numerical Mathematics, First ...

1.7.5 Upper bound for a sumSetwheref = ɛ 1 + 2ɛ 2 + 3ɛ 3 ,|ɛ i | ≤ ɛ, i = 1, 2, 3,and ɛ > 0 is a given number. Show thatSolution:|f| ≤ 6ɛ.|ɛ 1 + 2ɛ 2 + 3ɛ 3 | ≤ |ɛ 1 | + 2|ɛ 2 | + 3|ɛ 3 | ≤ 6ɛ.1.7.6 Local extrema of a functionPutf(x) = x(x 2 − 1), −2 ≤ x ≤ 2.Find all local extrema of f.Solution:Putf(x) = x(x 2 − 1), −2 ≤ x ≤ 2.Local extrema may occur at the endpoints x = −2 and x = 2 as well as thosepoints x satisfyingf ′ (x) = 0, and |x| < 2.We findf(−2) = (−2) · (4 − 1) = −6, f(2) = 2(4 − 1) = 6.f ′ (x) = x 2 − 1 + x · 2x = 3x 2 − 1.f ′ (x) = 0 for x = x 1 , x 2 ,wherex 1 = −1/ √ 3, x 2 = 1/ √ 3.We havef(x 1 ) = 2√ 39= 0.384900, f(x 2 ) = − 2√ 39= −0.384900,18