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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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// } class DefaultDemo { static void Main() { // Construct Test using a reference type Test<MyClass> x = new Test<MyClass>(); if(xobj == null) ConsoleWriteLine("xobj is null"); // Construct Test using a value type Test<int> y = new Test<int>(); if(yobj == 0) ConsoleWriteLine("yobj is 0"); } }

The output is shown here:

C# allows you to create generic structures The syntax is the same as for generic classes For example, in the following program, the XY structure, which stores X, Y coordinates, is generic:

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

// Demonstrate a generic struct using System; // This structure is generic struct XY<T> { T x; T y; public XY(T a, T b) { x = a; y = b; } public T X { get { return x; } set { x = value; } } public T Y { get { return y; }

Part I:

set { y = value; } } } class StructTest { static void Main() { XY<int> xy = new XY<int>(10, 20); XY<double> xy2 = new XY<double>(880, 990); ConsoleWriteLine(xyX + ", " + xyY); ConsoleWriteLine(xy2X + ", " + xy2Y); } }

The output is shown here:

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

10, 20 88, 99

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Like generic classes, generic structures can have constraints For example, this version of XY restricts type arguments to value types:

struct XY<T> where T : struct { //

We notice that k vanishes only when x = 2 or x = 4 These are the only candidates for local maxima or minima The second derivative is k ( x) = 6x 6 Now k ( 4) = 18 > 0, so x = 4 is the location of a local minimum Also k ( 2) = 18 < 0, so x = 2 is the location of a local maximum A glance at the graph of this function, as depicted in Figure 310, confirms our calculations