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Prove-that-even-obtaining-the-zero-s-the-following-equation-has-only-one-zero-f-t-1-2-t-1-t-2-t-2-t-2-

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Question Number 128896 by ZiYangLee last updated on 11/Jan/21
Prove that even obtaining the zero(s), the following equation has only one zero. f(t)=(1+(√2)t)(1−t^2 )+t^2 (t+(√2))
Provethatevenobtainingthezero(s),thefollowingequationhasonlyonezero.f(t)=(1+2t)(1−t2)+t2(t+2)
Answered by Olaf last updated on 11/Jan/21
f(t) = (1+(√2)t)(1−t^2 )+t^2 (t+(√2)) f′(t) = (√2)(1−t^2 )−2t(1+(√2)t)+2t(t+(√2))+t^2 f′(t) = 3((√2)−1)t^2 +2((√2)−1)t+(√2) Δ = 2^2 ((√2)−1)^2 −4×3×((√2)−1)(√2) = 4(√2)−12 < 0 The sign of f′ is the sign of 3((√2)−1) > 0 ⇒ f is a strictly increasing function and lim_(x→−∞) f(x) = lim_(x→−∞) t^3 = −∞ and lim_(x→+∞) f(x) = lim_(x→+∞) t^3 = +∞ ⇒ only one real root.
f(t)=(1+2t)(1−t2)+t2(t+2)f′(t)=2(1−t2)−2t(1+2t)+2t(t+2)+t2f′(t)=3(2−1)t2+2(2−1)t+2Δ=22(2−1)2−4×3×(2−1)2=42−12<0Thesignoff′isthesignof3(2−1)>0⇒fisastrictlyincreasingfunctionandlimx→−∞f(x)=limx→−∞t3=−∞andlimx→+∞f(x)=limx→+∞t3=+∞⇒onlyonerealroot.

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