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Numerade Educator

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Problem 59 Easy Difficulty

Show that if $ a \not= 0 $ and $ b \not= 0, $ then there exist numbers $ \alpha $ and $ \beta $ such that $ ae^x + be^{-x} $ equals either
$ \alpha \sinh (x + \beta) $ or $ \alpha \cosh (x + \beta) $
In other words, almost every function of the form $ f(x) = ae^e + be^{-x} $ is a shifted and stretched hyperbolic sine or cosine function.

Answer

$$\alpha=2 \sqrt{\pm a b}$$

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Video Transcript

okay. We know that we can write the derivative as to crow Sign acts over the square root of one plus four. Sign of axe. Therefore, at zero comma one, we have to over the spurt of one which is to therefore we can write this as one minus one equals two times X minus zero. You can read this in the formula. Y equals MX plus Pia's y equals two x plus one.