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# 16bf10lec2h - Math 16b Thomas Scanlon Autumn 2010 Thomas...

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Unformatted text preview: Math 16b Thomas Scanlon Autumn 2010 Thomas Scanlon Math 16b First derivative test for maxima/minima If f ( x ) is a function of one variable which has a relative maximum at a number a , then f ( a ) = 0. (Likewise, if f ( x ) has a relative minimum at a , then f ( a ) = 0.) Similarly, if F ( x , y ) is a function of two variables which has a relative extremum at some point ( a , b ) , then ∂ F ∂ x ( a , b ) = 0 and ∂ F ∂ y ( a , b ) = 0. Thomas Scanlon Math 16b An example Example Consider F ( x , y ) := x 2- 3 y 2 + xy +- x + 1 and nd where F may have a relative extremum. ∂ F ∂ x = 2 x + y- 1 ∂ F ∂ y =- 6 y + x Setting these equal to zero, we nd x = 6 y so that 13 y = 1 or y = 1 13 and x = 6 13 . Thomas Scanlon Math 16b Second derivative test Recall that if f ( x ) is a (su ciently di erentiable) function of a single variable and f ( a ) = 0 and f 00 ( a ) < 0, then f attains a local maximum at the point ( a , f ( a )) ....
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16bf10lec2h - Math 16b Thomas Scanlon Autumn 2010 Thomas...

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