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Application Of Derivatives

Question
CBSEENMA12035526
Wired Faculty App

A firm has found from experience that its profit as a function of x, the amount produced, is given by
straight p left parenthesis straight x right parenthesis space equals space minus straight x cubed over 3 plus 729 space straight x space minus 2500 comma space space space 0 space less or equal than space straight x space less or equal than space 35.
Find the value of x that maximises the profit.

Solution
straight p left parenthesis straight x right parenthesis space equals space straight x cubed over 3 plus 729 straight x space minus 2500 comma space space 0 less or equal than straight x less or equal than 35 space space space space rightwards double arrow space space space dp over dx space equals space minus straight x squared plus 729
Now,    dp over dx space equals space 0 space space gives space us space minus straight x squared plus 729 space equals space 0
therefore space space space straight x squared space equals space 729 comma space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space space space straight x space equals space 27 comma space space space minus 27 Now comma space space minus 27 space space not an element of space left square bracket 0 comma space 35 right square bracket comma space space space space space space space space space space space space space space space space rightwards double arrow space space space straight x space equals space 27 space space
                 fraction numerator straight d squared straight P over denominator dx squared end fraction space equals space minus 2 straight x
 When space straight x space equals space 27 comma space space space fraction numerator straight d squared straight p over denominator dx squared end fraction space equals space minus 54 space less than space 0
therefore space space space straight p space is space maximum space when space straight x space equals space 27.

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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