$\"\"$

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The range $\"\"$ of a projectile fired with an initial velocity $\"\"$ at an angle $\"\"$ with the horizontal is $\"\"$ , where $\"\"$ is the acceleration due to the gravity.

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Consider $\"\"$ .

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

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$\"\"$

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Equate derivative to zero.

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$\"\"$

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$\"\"$ , when $\"\"$ . $\"\"$

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Consider $\"\"$ .

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

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$\"\"$

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$\"\"$ .

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If $\"\"$ , then $\"\"$ .

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By second derivative test, the range is maximum at $\"\"$ .

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If $\"\"$ , then $\"\"$ .

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By second derivative test, the range is minimum at $\"\"$ .

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$\"\"$

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At $\"\"$ such that the range is maximum.