Application of Derivatives

Name: Wired Faculty - The Best Learning App
Rating: 4.6 (1864 reviews)

Question

Show that $straight y equals log space left parenthesis 1 plus straight x right parenthesis space minus space fraction numerator 2 straight x over denominator 2 plus straight x end fraction$ is an increasing function of x throughout its domain.

Solution

straight y space equals space log space left parenthesis 1 plus straight x right parenthesis space minus space fraction numerator 2 straight x over denominator 2 plus straight x end fraction therefore space space space space space dy over dx space equals space fraction numerator 1 over denominator 1 plus straight x end fraction minus fraction numerator left parenthesis 2 plus straight x right parenthesis. space 2 space minus space 2 space straight x. space 1 over denominator left parenthesis 2 plus straight x right parenthesis squared end fraction space equals space fraction numerator 1 over denominator 1 plus straight x end fraction space equals space fraction numerator 4 plus 2 straight x minus 2 straight x over denominator left parenthesis 2 plus straight x right parenthesis squared end fraction space equals fraction numerator 1 over denominator 1 plus straight x end fraction minus fraction numerator 4 over denominator left parenthesis 2 plus straight x right parenthesis squared end fraction space space space space space space space space space space space space space space space space space space space space equals fraction numerator left parenthesis 2 plus straight x right parenthesis squared minus 4 left parenthesis 1 plus straight x right parenthesis over denominator left parenthesis 1 plus straight x right parenthesis thin space left parenthesis 2 plus straight x right parenthesis squared end fraction space equals space fraction numerator 4 plus straight x squared plus 4 straight x minus 4 minus 4 straight x over denominator left parenthesis 1 plus straight x right parenthesis thin space left parenthesis 2 plus straight x right parenthesis squared end fraction space equals space fraction numerator straight x squared over denominator left parenthesis 1 plus straight x right parenthesis thin space left parenthesis 2 plus straight x right parenthesis squared end fraction greater than 0

open square brackets because space space space space straight x space equals space minus 1 close square brackets

therefore space space space space straight y space equals space log space left parenthesis 1 plus straight x right parenthesis space minus space fraction numerator 2 straight x over denominator 2 plus straight x end fraction space is space an space increasing space function space of space straight x space for all space straight x space space equals space minus 1.

For Daily Free Study Material Join wiredfaculty

Application Of Derivatives

Show that $straight y equals log space left parenthesis 1 plus straight x right parenthesis space minus space fraction numerator 2 straight x over denominator 2 plus straight x end fraction$ is an increasing function of x throughout its domain.

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.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction

For Daily Free Study Material Join wiredfaculty

Application Of Derivatives

Show that is an increasing function of x throughout its domain.

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.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Show that $straight y equals log space left parenthesis 1 plus straight x right parenthesis space minus space fraction numerator 2 straight x over denominator 2 plus straight x end fraction$ is an increasing function of x throughout its domain.