+91-8076753736[email protected]
logo
logo
-->

Application Of Derivatives

Question
CBSEENMA12035444
Wired Faculty App

Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
straight x cubed left parenthesis 2 straight x minus 1 right parenthesis cubed

Solution

Let  straight f left parenthesis straight x right parenthesis space equals space straight x cubed left parenthesis 2 straight x minus 1 right parenthesis cubed
therefore space space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space straight x cubed. space straight d over dx left square bracket left parenthesis 2 straight x minus 1 right parenthesis cubed right square bracket space plus space left parenthesis 2 straight x minus 1 right parenthesis cubed. space straight d over dx left parenthesis straight x cubed right parenthesis
                    equals straight x cubed. space 3 space left parenthesis 2 straight x minus 1 right parenthesis squared. space 2 space plus space left parenthesis 2 straight x minus 1 right parenthesis cubed. space 3 straight x squared space equals space 3 straight x squared space left parenthesis 2 straight x minus 1 right parenthesis squared space left parenthesis 2 straight x plus 2 straight x minus 1 right parenthesis space equals space 3 straight x squared left parenthesis 2 straight x minus 1 right parenthesis squared space left parenthesis 4 straight x minus 1 right parenthesis
straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space space space space space rightwards double arrow space space space space space space 3 straight x squared left parenthesis 2 straight x minus 1 right parenthesis squared space left parenthesis 4 straight x minus 1 right parenthesis space equals space 0 space space space space space space space space rightwards double arrow space space straight x space equals space 0 comma space space 1 half comma space space space 1 fourth. When space straight x less than 0 space space slightly comma space space space space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis thin space left parenthesis negative right parenthesis space equals space minus ve When space straight x greater than 0 space space slightly space space space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis space left parenthesis negative right parenthesis space equals space minus ve therefore space space space at space straight x space equals space 0 comma space space space straight f space apostrophe left parenthesis straight x right parenthesis space does space not space change space sign. therefore space space space space straight x space equals space 0 space is space straight a space point space of space straight f space inflexion. space
When space straight x less than 1 half space slightly comma space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis space equals space plus ve When space straight x greater than 1 half space slightly comma space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis space equals space plus ve therefore space space space at space space straight x space equals space 1 half comma space space space straight f space apostrophe left parenthesis straight x right parenthesis space does space not space change space sign. space
therefore space space space space straight x space equals space 1 half space is space straight a space point space of space inflexion. space
When space straight x less than 1 fourth space slightly comma space space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis thin space left parenthesis negative right parenthesis space equals space minus ve When space straight x greater than 1 fourth space slightly comma space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis space equals space plus ve therefore space space space at space straight x space equals space space 1 fourth comma space space space straight f space apostrophe left parenthesis straight x right parenthesis space changes space from space minus ve space to space plus ve therefore space space space straight f left parenthesis straight x right parenthesis space has space straight a space local space minimum space value space at space straight x space equals space 1 fourth and space this space value space equals space open parentheses 1 fourth close parentheses cubed space space space open parentheses 1 half minus 1 close parentheses cubed space equals space 1 over 64 cross times fraction numerator negative 1 over denominator 8 end fraction space equals space minus 1 over 512.

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

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation