Question 23: Which of the four functions listed below has no extremes?

The function \(y = \frac{{2x – 3}}{{x + 2}}\)

Defined set: \(D = \left( { – \infty ; – 2} \right) \cup \left( { – 2; + \infty } \right)\)

Yes \(y’ = \frac{7}{{{{\left( {x + 2} \right)}^2}}} > 0\,\forall x \in D \Rightarrow \) the function is always the same variable on each specified interval \( \Rightarrow \) function has no extremes.

Other functions can easily prove that y’ has solutions and change signs across the solutions. Particularly, the last function y’ is undefined at – 2, but the function is determined on and y’ changes sign to – 2, so there is a function with the extreme point x = – 2.

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