Prove that the function $ g(x) = \log x $ does not have any maximum or minimum.

Q: Prove that the function $ g(x) = \log x $ does not have any maximum or minimum.

Explanation: We have, \[ g(x) = \log x \Rightarrow g'(x) = \frac{1}{x} \] The function $ \log x $ is defined for $ x > 0 $, and for all such $ x $, \[ g'(x) = \frac{1}{x} > 0 \] So, $ g(x) $ is strictly increasing and $ g'(x) \neq 0 $ for any $ x > 0 $. Therefore, there does not exist any $ c \in (0, \infty) $ such that $ g'(c) = 0 $. Hence, the function $ g(x) = \log x $ does not have any maximum or minimum.

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Explanation: We have,
$$
g(x) = \log x \Rightarrow g'(x) = \frac{1}{x}
$$
The function $ \log x $ is defined for $ x > 0 $, and for all such $ x $,  
$$
g'(x) = \frac{1}{x} > 0
$$
So, $ g(x) $ is strictly increasing and $ g'(x) \neq 0 $ for any $ x > 0 $.  
Therefore, there does not exist any $ c \in (0, \infty) $ such that $ g'(c) = 0 $.  
Hence, the function $ g(x) = \log x $ does not have any maximum or minimum.

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