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:
$straight g left parenthesis straight x right parenthesis space equals straight x over 2 plus 2 over straight x comma space space space straight x greater than 0$

Solution

Let $straight g left parenthesis straight x right parenthesis space equals space straight x over 2 plus 2 over straight x comma space space space straight x greater than 0$
$straight g apostrophe left parenthesis straight x right parenthesis space equals space 1 half minus 2 over straight x squared$
$straight g apostrophe left parenthesis straight x right parenthesis space equals space 0 space space space space rightwards double arrow space space space space 1 half minus 2 over straight x squared space equals space 0 space space space space space rightwards double arrow space space space space straight x squared minus 4 space equals space 0 space space space space rightwards double arrow space space space space straight x space equals space 2 comma space minus 2$
$But space straight x space greater than 0 space space space left parenthesis given right parenthesis$
$therefore space space space straight x space equals space 2 space is space the space only space critical space point. space space space space space Now space space straight g apostrophe left parenthesis straight x right parenthesis space equals space fraction numerator straight x squared minus 4 over denominator 2 straight x squared end fraction space equals fraction numerator left parenthesis straight x minus 2 right parenthesis thin space left parenthesis straight x plus 2 right parenthesis over denominator 2 straight x squared end fraction space space space When space straight x less than 2 space space slightly comma space space straight g apostrophe left parenthesis straight x right parenthesis space equals space fraction numerator left parenthesis negative right parenthesis thin space left parenthesis plus right parenthesis over denominator plus end fraction space equals space minus ve space space space space When space straight x greater than 2 space slightly space straight g apostrophe left parenthesis straight x right parenthesis space equals space fraction numerator left parenthesis plus right parenthesis thin space left parenthesis plus right parenthesis over denominator plus end fraction space equals space plus ve space space space space space therefore space space space space at space straight x space equals space 2 comma space space straight g apostrophe left parenthesis straight x right parenthesis space changes space from space minus ve space to space space plus ve space space space space space therefore space straight g left parenthesis straight x right parenthesis space has space local space minimum space value space at space straight x space equals space 2 space and space space space space space local space minimum space value space at space straight x space equals 2 space and space space space space local space minimum space value space space equals space 2 over 2 plus 2 over 2 space equals space 1 plus 1 space equals space 2$

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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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