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How do you use the intermediate value theorem to show that there is a root of the equation #e^(-x^2)# over the interval (0,1)?

Answer 1

Cameron Alexander

You can't.

#e^(-x^2)# is not an equation, it is a function.

We can ask whether the function has a zero in #(0,1)#.

It does not. for any real #t#, e^t > 0#, so

#e^(-x^2) > 0# for all real #x#.

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Answer 2

Aurora Alvarado

To use the intermediate value theorem to show that there is a root of the equation ( e^{-x^2} ) over the interval (0,1), we need to demonstrate that the function changes sign within that interval.

Since ( e^{-x^2} ) is continuous on the interval (0,1) and ( e^{-x^2} ) is positive for ( x = 0 ) and ( e^{-x^2} ) is positive for ( x = 1 ), by the intermediate value theorem, there exists at least one ( c ) in the interval (0,1) such that ( e^{-c^2} = 0 ), implying that ( c ) is a root of the equation ( e^{-x^2} ) over the interval (0,1).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.