I want to ask  the quation that the second derivative test for maxima and minima as f ?(c)=0 then tangent is parallel to x axis at the point x=c .then how can we take double derivative at x=c that is f ??(c).>0 same thing about minima.

Asked by Kalyanrao Chavan | 22nd Aug, 2014, 10:18: PM

Expert Answer:

For space straight f apostrophe apostrophe left parenthesis straight c right parenthesis comma space straight f apostrophe left parenthesis straight c right parenthesis equals 0 Suppose space 0 less than straight f apostrophe apostrophe left parenthesis straight c right parenthesis rightwards double arrow 0 less than limit as straight h rightwards arrow 0 of fraction numerator straight f apostrophe left parenthesis straight c plus straight h right parenthesis minus straight f apostrophe left parenthesis straight c right parenthesis over denominator straight h end fraction rightwards double arrow 0 less than limit as straight h rightwards arrow 0 of fraction numerator straight f apostrophe left parenthesis straight c plus straight h right parenthesis minus 0 over denominator straight h end fraction rightwards double arrow limit as straight h rightwards arrow 0 of fraction numerator straight f apostrophe left parenthesis straight c plus straight h right parenthesis over denominator straight h end fraction greater than 0 Thus space for space straight h space sufficiently space small comma space fraction numerator straight f apostrophe left parenthesis straight c plus straight h right parenthesis over denominator straight h end fraction greater than 0 straight f apostrophe left parenthesis straight c plus straight h right parenthesis less than 0 comma space for space straight h less than 0 space left parenthesis straight f space is space decreasing space as space it space is space decreasing space from space the space left space from space straight x right parenthesis straight f apostrophe left parenthesis straight c plus straight h right parenthesis greater than 0 comma space for space straight h greater than 0 space left parenthesis straight f space is space increasing space as space it space space is space increasing space from space the space right space from space straight x right parenthesis By space first space derivative space test comma space straight f left parenthesis straight x right parenthesis space has space local space minimum space at space straight c.

Answered by Prasenjit Paul | 25th Aug, 2014, 11:30: AM