Question 20: Let the function \(y = f\left( x \right)\) be continuous on \(\mathbb{R}\) and have the derivative \(f’\left( x \right) = \frac {{\left( {x – 1} \right){{\left( {x – 2} \right)}^2}{{\left( {x – 3} \right)}^5}}}{ {\sqrt[3]{{x – 4}}}}\). How many extreme points does the function \(y = f\left( x \right)\) have?
We have \(f’\left( x \right) = 0 \Leftrightarrow \left[\begin{array}{l}x=1\\x=2\\x=3\end{array}\right\)[\begin{array}{l}x=1\x=2\x=3\end{array}\right\)
The table of signs of \(f’\left( x \right)\) is as follows:
Since \(f’\left( x \right)\) changes sign when x passes \(1,\;3,\;4\), the function \(y = f\left( x \right)\) has 3 extreme points.
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