Question

Manufacturer can sell x items at a price of rupees $open parentheses 5 minus straight x over 100 close parentheses space each.$ The cost price of x items is Rs. $open parentheses straight x over 5 plus 500 close parentheses.$ Find the number of items he should sell to earn maximum profit.

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Solution

straight C space equals space straight x over 5 plus 500 space straight R space equals space open parentheses 5 minus straight x over 100 close parentheses space left parenthesis straight x right parenthesis space equals space 5 straight x minus straight x squared over 100 straight P space equals space straight R space minus space straight C space equals space 5 straight x minus straight x squared over 100 minus straight x over 5 minus 500 therefore space space space space straight P space equals space 24 over 5 straight x minus straight x squared over 100 minus 500 space space space space space dP over dx space equals space 24 over 5 minus straight x over 50

dP over dx space equals 0 space space space space space rightwards double arrow space space space 24 over 5 minus straight x over 50 space equals space 0 space space space space space space rightwards double arrow space space space straight x over 50 space equals space 24 over 5

therefore space space space space space space space straight x space equals space 24 over 5 cross times 50 space equals space 240 space space space space fraction numerator straight d squared straight P over denominator dx squared end fraction space equals space minus 1 over 50

When space straight x space equals space 240 comma space space space space fraction numerator straight d squared straight P over denominator dx squared end fraction space equals space minus 1 over 50 less than 0 therefore space space space space straight P space is space maximum space when space straight x space equals space 240 therefore space space space space manufacturer space can space earn space maximum space profit comma space if space he space sells space 240 space mins. space

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