Finding Solutions Using the Intermediate Value Theorem

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Homework Statement



Show that the equation e^x = x + 2 had exactly two solutions. Explain your reasoning.

Homework Equations



Okay so I guess I'm using the intermediate value theorum and proving that f(a)f(b) < 0 we can find a c and d such that f(c) = 0 and f(d) = 0 so that by collorary it's true. This is an issue because I don't have boundaries for my function...

The Attempt at a Solution



create a function: f(x) = e^x - x - 2
and that's about as far as I got. I'm assuming I arbitarily select an interval and use that? Are there rule that I should follow though?
 
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I would believe you should select some values and check for a sign change...
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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