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First note that the function is undefined at x = 1 1 We calculate that h ( x) = ( x+1) 2 Thus the graph is everywhere increasing (except at x = 1) 2 We also calculate that h ( x) = ( x+1) 3 Hence h > 0 and the graph is concave up when x < 1 Likewise h < 0 and the graph is concave down when x > 1 Finally, as x tends to 1 from the left we notice that h tends to + and as x tends to 1 from the right we see that h tends to Putting all this information together, we obtain the graph shown in Figure 37

Related rate problems relate one rate, written as a derivative, to another rate written as a derivative An excellent example of a related rate problem, and one that is in nearly every calculus book including this one, is a ladder sliding down a wall (See Fig 6-1) The top of the ladder is moving down the wall while the bottom of the ladder is moving away fiom the wall The rate (speed) the top is moving down the wall can be related to the rate (speed) the bottom is moving away from the wall Thus the name of these problems, related rate problems

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We notice that k ( x) = 3x 2 + 6x 9 = 3( x 1) ( x + 3) So the sign of k changes at x = 1 and x = 3 We conclude that k is positive when x < 3; k is negative when 3 < x < 1; k is positive when x > 3 Finally, k ( x) = 6x + 6 Thus the graph is concave down when x < 1 and the graph is concave up when x > 1 Putting all this information together, and using the facts that k( x) when x and k( x) + when x + , we obtain the graph shown in Figure 38

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One of the great classical applications of the calculus is to determine the maxima and minima of functions Look at Figure 39 It shows some (local) maxima and (local) minima of the function f Notice that a maximum has the characteristic property that it looks like a hump: the function is increasing to the left of the hump and decreasing to the

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A little review is in order Related rate problems are similar to problems involving implicit differentiation Equations in the form y = f ( x ) such as y = x 2 + 2 x - 3 are differentiated term by term according to the rules for differentiating polynomials, products, quotients, or whatever Equations where the x's and y's are mixed together so the equation cannot be written as y = f ( x ) or x = f(y) (an x alone or a y alone on one side of the equation) are differentiated implicitly

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right of the hump The derivative at the hump is 0: the function neither increases nor decreases at a local maximum This is sometimes called Fermat s test Also, we see that the graph is concave down at a local maximum It is common to refer to the points where the derivative vanishes as critical points In some contexts, we will designate the endpoints of the domain of our function to be critical points as well Now look at a local minimum Notice that a minimum has the characteristic property that it looks like a valley: the function is decreasing to the left of the valley and increasing to the right of the valley The derivative at the valley is 0: the function neither increases nor decreases at a local minimum This is another manifestation of Fermat s test Also, we see that the graph is concave up at a local minimum Let us now apply these mathematical ideas to some concrete examples

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We begin by calculating the first derivative: k ( x) = 3x 2 6x 24 = 3( x + 2) ( x 4)

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