$\"\"$

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(a)

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The profit $\"\"$ for a company spending an amount $\"\"$ is $\"\"$ .

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Find the maximum profit differetiate $\"\"$ with respect to $\"\"$ .

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Consider $\"\"$ .

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$\"\"$

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Equate $\"\"$ to zero.

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$\"\"$

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The critical point is $\"\"$ .

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Consider $\"\"$ .

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Differetiate the above equation with respect to $\"\"$ .

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$\"\"$

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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By the second derivative test, $\"\"$ is negative so it is maximum.

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Thus, the maximum profit is $\"\"$ thousand dollars.

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$\"\"$

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(b)

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Find diminishing returns.

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Consider $\"\"$ .

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Find the diminsh, equate $\"\"$ to zero.

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$\"\"$

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The diminsh returns is $\"\"$ thousand dollars.

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$\"\"$

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(a) The maximum profit is $\"\"$ thousand dollars.

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(b) The diminsh returns is $\"\"$ thousand dollars.