Application of Derivatives

Question

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:
$sin to the power of 4 straight x plus cos to the power of 4 straight x comma space space space 0 less than straight x less than straight pi over 2$

Solution

Let $straight f left parenthesis straight x right parenthesis space equals space sin to the power of 4 straight x plus cos to the power of 4 straight x comma space space 0 less than straight x less than straight pi over 2$
$therefore space space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 4 space sin cubed straight x space cosx minus space 4 space cos cubed straight x space sinx space equals space 4 space sinx space cosx space left parenthesis sin squared straight x minus cos squared straight x right parenthesis$
$equals negative 2 left parenthesis 2 space sinx space cosx right parenthesis space left parenthesis cos squared straight x minus sin squared straight x right parenthesis space equals negative 2 space sin space 2 straight x space cos space 2 straight x space equals space minus sin space 4 straight x$
$straight f space apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space space space rightwards double arrow space space space minus sin space 4 straight x space equals space 0 space space space space space space space rightwards double arrow space space sin space 4 straight x space equals space 0$
$rightwards double arrow space space space space space 4 straight x space equals space straight pi comma space space 2 straight pi space space space space space rightwards double arrow space space space straight x space equals space straight pi over 4 comma space space straight pi over 2$
$When space space straight x less than straight pi over 4 space slightly comma space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus ve$
$When space straight x greater than straight pi over 4 space slightly comma space space straight f space apostrophe left parenthesis straight x right parenthesis space equals space plus ve$
$therefore space space space at space straight x space equals straight pi over 4 comma space space straight f apostrophe left parenthesis straight x right parenthesis space changes space from space minus ve space to space plus ve$
$therefore space space space space straight f space has space total space minimum space at space straight x space equals space straight pi over 4$
and local minimum value = $sin to the power of 4 straight pi over 4 plus cos to the power of 4 straight pi over 4 space equals space open parentheses fraction numerator 1 over denominator square root of 2 end fraction close parentheses to the power of 4 plus space open parentheses fraction numerator 1 over denominator square root of 2 end fraction close parentheses to the power of 4$
$equals space 1 fourth plus 1 fourth space equals space 1 half$
$Now comma space space straight x space equals straight pi over 2 space is space the space end space point When space straight x space less than space straight pi over 2 space slightly space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus left parenthesis negative ve right parenthesis space equals space plus ve therefore space space space straight f left parenthesis straight x right parenthesis space has space straight a space local space maxima space at space straight x space equals space straight pi over 2 and space this space local space maximum space value space space equals space sin to the power of 4 straight pi over 2 plus cos to the power of 4 straight pi over 2 space equals space 1 plus 0 space equals space 1$

For Daily Free Study Material Join wiredfaculty

Application Of Derivatives

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