2 Maximizing rectangle perimeters All rectangles with an area of 64 m have a perimeter given ? by P(? ) = 2 x + 128/x. ?wher? is the length of one. side of the rectangle. Find the absolute minimum value of the perimeter function. What are the Dimensions of the rectangle with minimum perimeter?

STEP_BY_STEP SOLUTION Step-1 Let f be a continuous function defined on an open interval containing a 1 number ‘c’.The number ‘c’ is critical value ( or critical number ). If f (c) 1 = 0 or f (c) is undefined. A critical point on that graph of f has the form (c,f(c)). Step-2 When an output value of a function is a maximum or a minimum over the entire domain of the function, the value is called the absolute maximum or the absolute minimum. Let f be a func tion with domain D and let c be a fixed constant in D . Then the output value f( ) is the 1. Absolute maximum value of f on D if and only if f(x) f(c) , for all x in D. 2. Absolute minimum value of f on D if and only if f(c) f(x) , for all x in D. Step_3 Given that ; x is the length of one side of the rectangle. Area of the rectangle is 64 m 2 128 Perimeter of the given rectangle is P(x) = 2x+ x .Clearly P(x) is a rational function and it is continuous for all x ,except zero. But here x is length of a rectangle , so it takes only positive values .Hence the function...