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f-x-x-2-2x-6-x-1-x-4-2x-3-2-x-lt-1-show-that-there-is-a-number-c-2-3-such-that-f-c-4-

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Question Number 90287 by jagoll last updated on 22/Apr/20
f(x) = { ((x^2 −2x+6 , x≥1)),((x^4 +2x^3 +2 ,x<1)) :} show that there is a number c ∈ (−2,3) such that f(c) = 4
f(x)={x2−2x+6,x⩾1x4+2x3+2,x<1showthatthereisanumberc∈(−2,3)suchthatf(c)=4
Answered by MJS last updated on 22/Apr/20
x^2 −2x+6=4 x^2 −2x+1=−1 (x−1)^2 =−1 x∉R x^4 +2x^3 +2=4 x^4 +2x^3 −2=0 g(x)=x^4 +2x^3 −2 g(−2)=−2<0 g(1)=1>0 ⇒ there must be at least one zero in [−2; 1]
x2−2x+6=4x2−2x+1=−1(x−1)2=−1x∉Rx4+2x3+2=4x4+2x3−2=0g(x)=x4+2x3−2g(−2)=−2<0g(1)=1>0⇒theremustbeatleastonezeroin[−2;1]

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