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Application Of Derivatives

Question

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Find the maximum and minimum values of $fraction numerator log space straight x over denominator straight x end fraction space in space 0 less than straight x less than infinity$ .

Solution

Let space straight y space equals space fraction numerator log space straight x over denominator straight x end fraction therefore space space space space dy over dx space equals space fraction numerator straight x. space begin display style 1 over straight x end style minus log space straight x. space 1 over denominator straight x squared end fraction space equals space fraction numerator 1 minus log space straight x over denominator straight x squared end fraction space space space space space space space space dy over dx space equals space 0 space gives space us space fraction numerator 1 minus logx over denominator straight x squared end fraction space equals space 0 rightwards double arrow space space space 1 minus logx space equals space 0 space space space space space rightwards double arrow space space space space logx space equals space 1 space equals space loge space space space space rightwards double arrow space space straight x space equals space straight e space space space space space space space space space space space space space dy squared over dx squared space equals space fraction numerator straight x squared open parentheses negative begin display style 1 over straight x end style close parentheses minus left parenthesis 1 minus log space straight x right parenthesis. space 2 straight x over denominator left parenthesis straight x squared right parenthesis squared end fraction space equals space fraction numerator negative straight x minus 2 straight x plus 2 straight x space logx over denominator straight x to the power of 4 end fraction space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator negative 3 straight x plus 2 straight x space logx over denominator straight x to the power of 4 end fraction space equals space fraction numerator negative 3 plus 2 space log space straight x over denominator straight x cubed end fraction At space straight x space equals space straight e comma space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction space equals space fraction numerator negative 3 plus 2 space log space over denominator straight e cubed end fraction space equals space fraction numerator negative 3 plus 2 over denominator straight e cubed end fraction space equals space minus 1 over straight e cubed less than 0 space space space space space space space space space

therefore space space space space straight y space is space maximum space when space straight x space equals space straight e space and space maximum space value space space equals space fraction numerator log space straight e over denominator straight e end fraction space equals space 1 over straight e

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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