Problem

Prove that the function f(x)=x2−4x+4 has at least one root in the interval [1,2].

Answer

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Answer

Step 4: Since f(1)>0 and f(2)=0, by the Intermediate Value Theorem, there exists at least one c in [1,2] such that f(c)=0. Therefore, the function f(x)=x2−4x+4 has at least one root in the interval [1,2].

Steps

Step 1 ：Step 1: We first calculate the function values at the endpoints of the interval, i.e., f(1) and f(2).

Step 2 ：Step 2: f(1)=(1)2−4(1)+4=1−4+4=1

Step 3 ：Step 3: f(2)=(2)2−4(2)+4=4−8+4=0

Step 4 ：Step 4: Since f(1)>0 and f(2)=0, by the Intermediate Value Theorem, there exists at least one c in [1,2] such that f(c)=0. Therefore, the function f(x)=x2−4x+4 has at least one root in the interval [1,2].

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