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Application Of Derivatives

Question
CBSEENMA12035436
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:
sin 2 straight x minus straight x comma space space minus straight pi over 2 less or equal than straight x less or equal than straight pi over 2





Solution
Let space straight f left parenthesis straight x right parenthesis space equals space sin space 2 straight x space minus straight x comma space space space minus straight pi over 2 less or equal than straight x less or equal than straight pi over 2 therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 2 space cos space 2 straight x minus 1 space space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space space rightwards double arrow space space space space 2 space cos space 2 straight x space minus 1 space space equals space 0 space space space space space rightwards double arrow space space space cos space 2 straight x space equals space 1 half rightwards double arrow space space space space cos space 2 straight x space equals space cos space straight pi over 3 comma space space cos space open parentheses negative straight pi over 3 close parentheses therefore space space space space 2 straight x space equals space straight pi over 3 comma space space space space fraction numerator negative straight pi over denominator 3 end fraction space space space space space space space space rightwards double arrow space space space space straight x space equals space straight pi over 6 comma space space space minus straight pi over 6 When space straight x less than negative straight pi over 6 space slightly comma space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus ve When space straight x space greater than negative straight pi over 6 space slightly comma space straight f apostrophe left parenthesis straight x right parenthesis space equals space plus ve therefore space space space at space straight x space equals space minus straight pi over 6 comma space space space straight f apostrophe left parenthesis straight x right parenthesis space changes space from space minus ve space to space ve therefore space space space straight f left parenthesis straight x right parenthesis space has space straight a space local space minima space at space straight x space equals space minus straight pi over 6
and space this space local space minimum space value space space equals space sin space open parentheses negative straight pi over 3 close parentheses minus open parentheses negative straight pi over 6 close parentheses space equals space minus sin space straight pi over 3 space plus space straight pi over 6 space equals negative fraction numerator square root of 3 over denominator 2 end fraction plus straight pi over 6
When space straight x less than straight pi over 6 space slightly comma space straight f apostrophe left parenthesis straight x right parenthesis space equals space plus ve When space straight x greater than straight pi over 6 space slightly comma space space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus ve therefore space space space at space straight x space equals space straight pi over 6 comma space space space straight f left parenthesis straight x right parenthesis space changes space from space plus ve space to space minus ve therefore space space space straight f left parenthesis straight x right parenthesis space has space local space maxima space at space straight x space equals space straight pi over 6 and space this space local space maximum space value space space equals space sin space straight pi over 3 minus straight pi over 6 space equals space fraction numerator square root of 3 over denominator 2 end fraction minus straight pi over 6

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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