121616949-math.123 - 6.1 Optimization 109 whether this is...

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6.1 Optimization 109 whether this is true. Since the left side is clearly larger than 4 · 4 which is clearly larger than 3, this settles the question.) A similar analysis shows that there is also no global minimum. The graph of f ( x ) on ( - 2 , 2) is shown in figure 6.2 . EXAMPLE 6.7 Of all rectangles of area 100, which has the smallest perimeter? First we must translate this into a purely mathematical problem in which we want to find the minimum value of a function. If x denotes one of the sides of the rectangle, then the adjacent side must be 100 /x (in order that the area be 100). So the function we want to minimize is f ( x ) = 2 x + 2 100 x since the perimeter is twice the length plus twice the width of the rectangle. Not all values of x make sense in this problem: lengths of sides of rectangles must be positive, so x > 0. If x > 0 then so is 100 /x , so we need no second condition on x . We next find f ( x ) and set it equal to zero: 0 = f ( x ) = 2 - 200 /x 2 . Solving f ( x ) = 0 for x gives us x = ± 10. We are interested only in x > 0, so only the value x = 10 is of interest. Since
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