Prove that the function $ f(x) = e^x $ does not have any maximum or minimum.

Q: Prove that the function $ f(x) = e^x $ does not have any maximum or minimum.

Explanation: We have, \[ f(x) = e^x \Rightarrow f'(x) = e^x \] Since $ e^x > 0 $ for all $ x \in \mathbb{R} $, $ f'(x) \neq 0 $ for any real $ x $. Therefore, there is no critical point $ c \in \mathbb{R} $ such that $ f'(c) = 0 $. Hence, the function $ f(x) = e^x $ does not have any maximum or minimum.

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Explanation: We have,
$$
f(x) = e^x \Rightarrow f'(x) = e^x
$$
Since $ e^x > 0 $ for all $ x \in \mathbb{R} $, $ f'(x) \neq 0 $ for any real $ x $.  
Therefore, there is no critical point $ c \in \mathbb{R} $ such that $ f'(c) = 0 $.  
Hence, the function $ f(x) = e^x $ does not have any maximum or minimum.

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