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(a)

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The functions $\"\"$ and $\"\"$ are positive, increasing and concave up on the interval $\"\"$ .

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Therefore $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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Consider $\"\"$ .

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Apply derivative on each side.

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$\"\"$

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Again apply derivative on each side.

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$\"\"$

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$\"\"$ , since $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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From the concavity test, $\"\"$ is a concave up over the interval $\"\"$ .

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(b)

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The functions $\"\"$ and $\"\"$ are decreasing on the interval $\"\"$ .

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

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Consider $\"\"$ .

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Apply derivative on each side.

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$\"\"$

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Again apply derivative on each side.

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$\"\"$

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$\"\"$ , since $\"\"$ , $\"\"$ and $\"\"$ .

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From the concavity test, $\"\"$ is a concave up over the interval $\"\"$ .

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(c)

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The function $\"\"$ is increasing and $\"\"$ is decreasing on the interval $\"\"$ .

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

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Consider $\"\"$ .

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$\"\"$

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$\"\"$ is either concave up or down, since $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$ is a linear over the interval $\"\"$ .

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Example :

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(1)

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Consider the function are $\"\"$ and $\"\"$ on $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ is a concave up over the interval $\"\"$ .

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(2)

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Consider the function are $\"\"$ and $\"\"$ on $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ is a concave down over the interval $\"\"$ .

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(3)

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Consider the function are $\"\"$ and $\"\"$ on $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ is a linear over the interval $\"\"$ .

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(a) $\"\"$ is a concave up on $\"\"$ when $\"\"$ and $\"\"$ are positive, increasing and concave up on $\"\"$ .

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(b) $\"\"$ is a concave up on $\"\"$ when $\"\"$ and $\"\"$ are decreasing on $\"\"$ .

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(c) $\"\"$ is a linear on $\"\"$ when $\"\"$ is increasing and $\"\"$ is decreasing on $\"\"$ .