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Let f(x) be a continuous function, defined for all real numbers, with f(1)=6.27, f(3)=−2.14, and f(7)=16.49. How many zeros must the function have?




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F(1) = 6.27
f(3) = -2.14
f(7) = 16.49

Notice that the function is changing sign between x=1 and x=3, so there must exist a zero in the interval [1,3] by intermediate value theorem.

For the same reason there must exist another zero in the interval [3, 7].

That means f(x) must have at least two zeroes.

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