Question

Use the intermediate value theorem to prove that there is a positive number c such that c2 = 2.

Answer 1

So lets try to prove it,

So let's consider the function f(x) = x^2.

Since f(x) is a polynomial, then it is continuous on the interval (- infinity, + infinity).

Using the Intermediate Value Theorem,

it would be enough to show that at some point a f(x) is less than 2 and at some point b f(x) is greater than 2. For example, let a = 0 and b = 3.

Therefore, f(0) = 0, which is less than 2, and f(3) = 9, which is greater than 2. Applying IVT to f(x) = x^2 on the interval [0,3}.

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