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A function $\"\"$ has a local maximum at $\"\"$ if there is an open interval $\"\"$ containing $\"\"$ so that for all $\"\"$ in $\"\"$ , $\"\"$ , then $\"\"$ is called as a local maximum value of $\"\"$ .

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A function $\"\"$ has a local minimum at $\"\"$ if there is an open interval $\"\"$ containing $\"\"$ so that for all $\"\"$ in $\"\"$ , $\"\"$ , then $\"\"$ is called as a local minimum value of $\"\"$ .

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The graph of the function $\"\"$ on the interval $\"\"$ is :

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$\"\"$

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Observe the above graph :

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$\"\"$ has a local maximum at $\"\"$ and $\"\"$ , because $\"\"$ and $\"\"$ .

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The local maxima : $\"\"$ , $\"\"$ .

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$\"\"$ has a local minimum at $\"\"$ , because $\"\"$ .

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The local minimum : $\"\"$ , $\"\"$ .

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The function is increasing for all values of $\"\"$ between $\"\"$ to $\"\"$ and $\"\"$ to $\"\"$ .

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Thus, the function is increasing over the intervals $\"\"$ and $\"\"$ .

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The function is decreasing for all values of $\"\"$ between $\"\"$ to $\"\"$ and $\"\"$ to $\"\"$ .

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Thus, the function is decreasing over the intervals $\"\"$ and $\"\"$ .

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The graph of the function $\"\"$ on the interval $\"\"$ is :

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$\"\"$

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The local maxima : $\"\"$ , $\"\"$ .

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The local minimum : $\"\"$ , $\"\"$ .

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Increasing over the intervals $\"\"$ and $\"\"$ .

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Decreasing over the intervals $\"\"$ and $\"\"$ .