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Question Number 139190 by mathdanisur last updated on 23/Apr/21
a;b∈R , ((∣ax+b∣)/(1+x^2 )) ≤ 1 , ∀x∈R prove: ∣a∣≤2 ; ∣b∣≤1
a;b∈R,∣ax+b∣1+x2⩽1,∀x∈Rprove:∣a∣⩽2;∣b∣⩽1
Answered by mitica last updated on 24/Apr/21
x=0⇒∣b∣≤1 x=1⇒∣a+b∣≤2 x=−1⇒∣a−b∣≤2 ∣a∣=∣((2a)/2)∣=((∣a+b+a−b∣)/2)≤((∣a+b∣+∣a−b∣)/2)≤((2+2)/2)=2
x=0⇒∣b∣⩽1x=1⇒∣a+b∣⩽2x=−1⇒∣a−b∣⩽2∣a∣=∣2a2∣=∣a+b+a−b∣2⩽∣a+b∣+∣a−b∣2⩽2+22=2
Commented by mathdanisur last updated on 24/Apr/21
thank you sir, but ∣b∣≤1? please
thankyousir,but∣b∣⩽1?please
Commented by mitica last updated on 24/Apr/21
x=0⇒((∣a∙0+b∣)/(0^2 +1))≤1⇒((∣b∣)/1)≤1⇒∣b∣≤1
x=0⇒∣a⋅0+b∣02+1⩽1⇒∣b∣1⩽1⇒∣b∣⩽1
Answered by mr W last updated on 24/Apr/21
Commented by mr W last updated on 24/Apr/21
let′s generally look at the function y=x^2 +bx+c there are three cases. case 1: y>0 for x∈R x^2 +bx+c=0 has no real root, i.e. Δ=b^2 −4c<0 case 2: y≥0 for x∈R x^2 +bx+c=0 has one double real root, i.e. Δ=b^2 −4c=0 case 3: x^2 +bx+c=0 has two real rootx, i.e. Δ=b^2 −4c>0 y>0 if x<x_1 or x>x_2 y<0 if x_1 <x<x_2 y=0 if x=x_1 or x_2 ⇒such that x^2 +bx+c≥0 for x∈R, Δ=b^2 −4c≤0
let′sgenerallylookatthefunctiony=x2+bx+ctherearethreecases.case1:y>0forx∈Rx2+bx+c=0hasnorealroot,i.e.Δ=b2−4c<0case2:y⩾0forx∈Rx2+bx+c=0hasonedoublerealroot,i.e.Δ=b2−4c=0case3:x2+bx+c=0hastworealrootx,i.e.Δ=b2−4c>0y>0ifx<x1orx>x2y<0ifx1<x<x2y=0ifx=x1orx2⇒suchthatx2+bx+c⩾0forx∈R,Δ=b2−4c⩽0
Commented by mr W last updated on 24/Apr/21
((∣ax+b∣)/(1+x^2 ))≤1 for x∈R ⇒ −1≤((ax+b)/(1+x^2 ))≤1 ⇒ −1−x^2 ≤ax+b≤1+x^2 1) ax+b≤1+x^2 x^2 −ax+1−b≥0 Δ=a^2 +4(b−1)≤0 ...(i) 4(b−1)≤0 ⇒b≤1 2) −1−x^2 ≤ax+b x^2 +ax+b+1≥0 Δ=a^2 −4(b+1)≤0 ...(ii) 4(b+1)≥a^2 ≥0 ⇒b≥−1 ⇒−1≤b≤1 ⇒∣b∣≤1 (i)+(ii): 2a^2 −4−4≤0 a^2 ≤4 ⇒∣a∣≤2
∣ax+b∣1+x2⩽1forx∈R⇒−1⩽ax+b1+x2⩽1⇒−1−x2⩽ax+b⩽1+x21)ax+b⩽1+x2x2−ax+1−b⩾0Δ=a2+4(b−1)⩽0…(i)4(b−1)⩽0⇒b⩽12)−1−x2⩽ax+bx2+ax+b+1⩾0Δ=a2−4(b+1)⩽0…(ii)4(b+1)⩾a2⩾0⇒b⩾−1⇒−1⩽b⩽1⇒∣b∣⩽1(i)+(ii):2a2−4−4⩽0a2⩽4⇒∣a∣⩽2
Commented by mathdanisur last updated on 24/Apr/21
Perfect Sir thanks.. How can this be Sir pliase.. if: a;b∈R, ∣ax^3 +bx∣≤1, ∀∣x∣≤1 prov: ∣bx^3 +ax∣≤1, ∣3ax^2 +b∣≤9, ∀∣x∣≤1
PerfectSirthanks..HowcanthisbeSirpliase..if:a;b∈R,∣ax3+bx∣⩽1,∀∣x∣⩽1prov:∣bx3+ax∣⩽1,∣3ax2+b∣⩽9,∀∣x∣⩽1
Commented by mr W last updated on 24/Apr/21
please open a new thread for this new question!
pleaseopenanewthreadforthisnewquestion!

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