Question

Find the local maxima or local minima, if any, of the following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
$straight h left parenthesis straight x right parenthesis space equals space sinx plus cosx comma space 0 space less than space straight x space less than space straight pi over 2$

Solution

Let $straight f left parenthesis straight x right parenthesis space equals space sinx plus cosx space comma space 0 space space less than space straight x less than straight pi over 2$
$therefore space space space straight f apostrophe left parenthesis straight x right parenthesis space equals space cosx minus space sinx$
Now, $straight f apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space rightwards double arrow space space space space cos space straight x minus sin space straight x space equals space 0 space space space space rightwards double arrow space space space sinx space equals space cosx space space rightwards double arrow space space space tanx space equals space 1 space space space rightwards double arrow space space straight x space equals space straight pi over 4$
Now,
$straight f apostrophe left parenthesis straight x right parenthesis space equals space square root of 2 space open parentheses fraction numerator 1 over denominator square root of 2 end fraction cosx space minus fraction numerator 1 over denominator square root of 2 end fraction sinx close parentheses space equals space square root of 2 space open parentheses cosx space cos straight pi over 4 minus sinx space sin straight pi over 4 close parentheses space space space space space space space space space space equals square root of 2 space cos space open parentheses straight x plus straight pi over 4 close parentheses When space straight x less than straight pi over 4 space slightly comma space space straight f apostrophe left parenthesis straight x right parenthesis space equals space plus ve When space straight x greater than space straight pi over 4 space slightly comma space straight f apostrophe left parenthesis straight x right parenthesis space equals space minus ve therefore space space space space at space straight x space equals space straight pi over 4 comma space space straight f apostrophe left parenthesis straight x right parenthesis space changes space from space plus ve space to space minus ve therefore space space space straight f left parenthesis straight x right parenthesis space has space local space maxima space at space straight x space equals space straight pi over 4 and space local space maximum space value space space equals space sin space space straight pi over 4 plus space cos space straight pi over 4 space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator square root of 2 end fraction plus fraction numerator 1 over denominator square root of 2 end fraction space equals space fraction numerator 2 over denominator square root of 2 end fraction space equals space square root of 2$

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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.

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