Show how to use the Intermediate Value Theorem to show that the equation $\displaystyle {x}^{{2}}-\sqrt{{{2}{x}}}={1}$ has a solution between 0 and 2.

Let $\displaystyle {f{{\left({x}\right)}}}={x}^{{2}}-\sqrt{{{2}{x}}}$. In order for the Intermediate Value Theorem to apply we must first check that $\displaystyle {f}$ is Select an answer a polynomial a rational function solvable differentiable continuous on the interval [0,2]. You should verify this and be able to explain why it is the case.

We check the value of $\displaystyle {f}$ at the left endpoint of the interval [0,2]:

$\displaystyle {f}$() = .

And we check the value of $\displaystyle {f}$ at the right endpoint of the interval [0,2]:

$\displaystyle {f}$() = .

The Intermediate Value Theorem guarantees a solution to $\displaystyle {f{{\left({x}\right)}}}={1}$ in the interval (0,2) because

$\displaystyle {f{{(}}}$$\displaystyle {)}\le$ $\displaystyle \le{f{{(}}}$$\displaystyle {)}$.