Relative Maxima and Minima sections 4.3

Transcription

1 Relative Maxima and Minima setions 4.3 Definition. By a ritial point of a funtion f we mean a point x 0 in the domain at whih either the derivative is zero or it does not exists. So, geometrially, one of the following ases happen at a ritial point: (i) the tangent line is horizontal (ii) the tangent line is vertial (this orresponds to the ase where the derivative is one of ± ) (iii) the tangent line does not exist ; there is a usp on the graph at x 0 Example. Find the ritial points of the funtion f(x) = x x 3 1 Solution. f (x) = (x ) (x 3 1) (x )(x 3 1) (x 3 1) = = x(x3 + ) (x 3 1) Now we need to look for the points at whih the derivative is zero or the derivative does not exist. f (x) = 0 x = 0, 3 there is no point in the domain where f is not differentiable (note that the point x = 1 is not in the domain at all, so it is not onsidered as a ritial point). So, the only ritial points are x = 0, 3. Definition. A funtion f is said to have minimum (loal minimum) at x 0 if there exists some open interval I ontaining x 0 suh that f(x 0 ) f(x) for all x I Definition. A funtion f is said to have there exists some open interval I ontaining x 0 suh that maximum (loal maximum) at x 0 if f(x) f(x 0 ) for all x I 1

2 Question. The following theorem tells us where to searh for extrema ; in fat it says that we need to llok for the ritial points if we really want to searh for the extrema. Theorem. If a funtion f has a extrema over an interval I at a point x 0 I and if x 0 is not an endpoint of I, then x 0 is a ritial point of f, and therefore either f(x 0 ) = 0 or f (x 0 ) does not exist. First-Derivative Test for Relative Extrema. Suppose that is a ritial point of a ontinuous funtion f (f () may or may not exist). (i) If on both sides of we have this situation: then f has a minimum at. f + f (ii) If on both sides of we have this situation: then f has a maximum at. f + f (iii) If on both sides of the derivative f does not hange sign: f + + f f f

3 then f has neither a max or min at. Example (from the textbook). Find the maximums and minimums of the funtion f(x) = x 5 3 x 3 = 3 x 5 3 x Solution. f (x) = 5 3 x 3 3 x 1 3 = 5 3 x x = 5( 3 x )( 3 x) 3 3 x = 5x 3 3 x f (x) = 0 5x = 0 x = 5 (a ritial point) The funtion is not differentiable at the point x = 0 of the domain (another ritial point). So we have only two ritial points: x = 0 and x = 5 3 x + f 5x f max min Seond-Derivative Test for Relative Extrema. Suppose f (x) is ontinuous on an interval ontaining a point and that is a ritial point of type f () = 0. Then 3

4 If f () > 0, then is a point of minimum for f. If f () < 0, then is a point of maximum for f. If f () = 0, then no onlusion an be made about. Note. The first-derivative test is used for both types of ritial points no matter whether f () = 0 or f () does not exist. But, the seond-derivative test is used only for the ritial points satisfying f () = 0. For example, in the previous example, the seond-derivative test annot be applied for the ritial point x = 0. But, it an be used to deide about x = 5. f (x) = 1 3 (5x )x 1 3 f (x) = 1 3 (5x ) (x 1 3 ) (5x )(x 1 3 ) = (5)(x 1 3 ) + 3 (5x )( x 4 3 ) = 5x 3 3 x 9x 3 x = 15x (5x ) = 10x + 9x 3 x 9x 3 x f ( 5 ) > 0 5 gives minimum Example (setion 4.3 exerise 19). Find the the ritial points of the funtion f(x) = (x + ) 3 (x 4) 3, and determine whih ritial points give maxima and whih ones give minima. 4

5 Solution. f (x) = {(x + ) 3 } {(x 4) 3 } + {(x + ) 3 }{(x 4) 3 } = {3(x + ) }{(x 4) 3 } + {(x + ) 3 }{3(x 4) } = 3(x + ) (x 4) {(x 4) + (x + )} = 3(x + ) (x 4) {x } = 6(x + ) (x 4) (x 1) f (x) = 0 x =, 1, 4 (the only ritial points) (there are no ritial points at whih f does not exist). The only term whose sign must be determined is the term (x 1) beause the signs of the other terms don t hange. x 1 f f min 5

6 Note. The seond-derivative test annot deide about the ritial points - and 4 beause at these points we have f (x) = 0. But for the ritial point x = 1 we an use that test: f (x) = 6{(x+) } {(x 4) }{(x 1)}+6{(x+) }{(x 4) } {(x 1)}+6{(x+) }{(x 4) }{(x 1)} = 6{(x + )}{(x 4) }{(x 1)} + 6{(x + ) }{(x 4)}{(x 1)} + 6{(x + ) }{(x 4) }{1}+ = 6(x + )(x 4){(x 5x + 4) + (x + x ) + (x x 8)} = = 6(x + )(x 4){5x 10x 4} f (1) = (6)(3)( 3)( 9) > 0 x = 1 is a minimum Note. It seems that the first-derivative test is easier and more omprehensive than the seondderivative test, therefore in your exam use the first-derivative test. Example. Find the ritial points of the funtion f(x) = 3x 4 + 8x 3 + 6x and determine whih ones give maxima and whih ones give minima. Solution. f (x) = 1x 3 + 4x + 1x = 1x(x + x + 1) To fatorize x + x + 1 into linear forms, we do this: x + x + 1 = 0 x = b± b 4a a So then f (x) = 1x(x 1) = ± 4 4 = +±0 = 1 (multiple root) We only need to determine the sign of x beause the sign of the other term does not hange. 6

7 x + f + f 1 0 minimum 7