$\"\"$

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Critical number :

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A critical number of a function $\"\"$ is a number $\"\"$ in the domain of $\"\"$ such that either $\"\"$ or $\"\"$ does not exist.

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$\"\"$

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The function $\"\"$ .

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Solutions of $\"\"$ and points where $\"\"$ does not exist are the critical numbers.

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Differentiate on each side with respect to $\"image\"$ .

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$\"\"$

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Apply quotient rule in derivatives $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$

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Find the critical numbers by equating the first derivative to zero.

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$\"\"$

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$\"\"$

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$\"\"$

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Apply zero product property.

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$\"\"$ and $\"\"$

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$\"\"$ and $\"\"$ .

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$\"\"$

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The function $\"\"$ does not exist when $\"\"$ .

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$\"\"$

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The discriminant of the above equation: $\"\"$ .

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Since $\"\"$ , the roots are imaginary.

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Hence, they are not considered.

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Critical numbers are $\"\"$ and $\"\"$ .

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$\"\"$

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Critical numbers are $\"\"$ and $\"\"$ .