A root of f(x) = 0 lies between x = a and x = b if a.) f(a)f(b)= 0 b.) f(a)f(b)>0 c.) f(a)f(b) <0 d.) None of these
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Assuming that f(x) is a continuous function (naively the function has no holes, jumps, and the function does not tend to positive or negative infinity -- i.e. you could draw the function from x=a to x=b without lifting your pencil) then the answer is (c).
If f(a)f(b)<0 then exactly one of f(a) or f(b) is negative and the other is positive. That means that the function changes sign on the interval, and so must pass through zero.
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** If we do not know that f(x) is continuous then the answer is (d). None of the conditions guarantees a zero on the interval from a to b. Consider the function `f(x)=1/x` from -2 to 2. `f(-2)=-1/2,f(2)=1/2` so f(a)f(b)<0 but there is no x in the interval from -2 to 2 where the function is 0.
Consider ``f(x)=1 for x<0,f(x)=-1 for x`>=` 0. Again f(-2)=1 and f(2)=-1 but there is no x such that f(x)=0.
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