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$\"\"$

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(a)

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The function is $\"\"$ .

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Graph :

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Graph the function $\"\"$ :

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$\"\"$

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Observe the graph :

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Concave upward :

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The graph of $\"\"$ lies below all of its tangent to the graph of the curve at a point.

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The function $\"\"$ is concave upward over the intervals $\"\"$ , $\"\"$ .

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Concave downward :

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The graph of $\"\"$ lies above all of its tangent to the graph of the curve at a point.

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The function $\"\"$ is concave downward over the intervals $\"\"$ and $\"\"$ .

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Point of inflection :

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Inflection points are the points at which the concavity changes from up to down or down to up.

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The inflection points occurs at $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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The coordinates of the inflection points are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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(b)

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The function is $\"\"$ .

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

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$\"\"$

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$\"\"$ .

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Again differentiate on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$ .

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Graph :

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Graph the function $\"\"$ : $\"\"$

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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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Observe the garph :

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$\"\"$ over the intervals $\"\"$ and $\"\"$ , therefore the function $\"\"$ is concave upward on the intervals $\"\"$ and $\"\"$ .

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$\"\"$ over the intervals $\"\"$ and $\"\"$ , therefore the function $\"\"$ is concave downward on the intervals $\"\"$ and $\"\"$ .

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Inflection points :

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Inflection points are the points at which the concavity changes from up to down or down to up.

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Where $\"\"$ are the inflection points.

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The inflection points occurs at $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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The coordinates of the inflection points are $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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(a)

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Concave up over the intervals $\"\"$ and $\"\"$ .

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Concave down over the intervals $\"\"$ and $\"\"$ .

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

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(b)

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concave up over the intervals $\"\"$ and $\"\"$ .

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concave down over the intervals $\"\"$ and $\"\"$ .

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