$\"\"$

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The condition are :

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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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I/D test :

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If $\"\"$ on the interval, then $\"\"$ is increasing on the interval.

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If $\"\"$ on the interval, then $\"\"$ is decreasing on the interval.

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Concavity test :

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If $\"\"$ for all $\"\"$ in the interval, then the graph of $\"\"$ is concave upward on the interval.

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If $\"\"$ for all $\"\"$ in the interval, then the graph of $\"\"$ is concave downward on the interval.

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Graph :

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Graph the function such that all the conditions are satisfied :

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Using above tests,

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The critical points are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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Therefore it has a either local maximum or minimum at the critical points.

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$\"\"$ represents, the inflection point is $\"\"$ .

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$\"\"$ represents that $\"\"$ is increasing on $\"\"$ .

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$\"\"$ represents that $\"\"$ is decreasing on $\"\"$ .

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$\"\"$ represents that $\"\"$ is concave upward on $\"\"$ .

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$\"\"$ represents that $\"\"$ is concave downward on $\"\"$ .

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$\"\"$ represents that $\"\"$ is an even function.

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$\"\"$

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$\"\"$

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Graph the function is drawn such that all the conditions are satisfied :

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$\"\"$ .