sol14-1 - SOLUTIONS CHAPTER 14.1 MATH 132 WI01 2. Function...

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SOLUTIONS CHAPTER 14.1 MATH 132 WI01 2. Function is 00 00 decreasing in ( -∞ , - 1), then 00 % 00 increasing in ( - 1,0), again 00 00 decreasing in (0,1), and finally 00 % 00 increasing in (1, ). Relative extrema are points where the function’s behaviour changes from increasing to de- creasing - maximum - which in our case is (0 , 0), or from decreasing to increasing - minimum - which in our case are ( - 1 , - 1) and (1 , - 1). 6. What we are interested in is the sign of the derivative, derivative which is already given in our case. Hence, let’s look for sign of f 0 ( x ) = 2 x ( x - 1) 3 . f 0 ( x ) equals zero in x = 0 and in x = 1, so we have to check sign between these values, and outside these values: plug in - 1, we get 2( - 1)( - 2) 3 = 16, hence positive, hence the original f is increasing in the interval ( -∞ , 0); plug in 1 2 , we get 2 * ( 1 2 )( 1 2 - 1) 3 = ( - 1 2 ) 3 = - 1 8 , hence negative, so the original f function will be decreasing in the interval (0 , 1); plug in 2, we get 2 * 2(2 - 1) 3 = 4, hence again positive, so now the function f is again increasing in the interval (1 , ). As you can see, at 0 we have a change from increasing to decreasing, so it’s a relative
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sol14-1 - SOLUTIONS CHAPTER 14.1 MATH 132 WI01 2. Function...

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