This problem illustrates the Envelope Theorem, whichrelates the maxima of z = f(x, y)

Chapter 15, Problem 48

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This problem illustrates the Envelope Theorem, whichrelates the maxima of z = f(x, y) subject to the constraintx = c to the contour diagram in Figure 15.37 andthe cross-sections in Figure 15.38.(a) For each value c, there is a maximum value off(x, y) with x = c. On Figure 15.37, sketch thecurve that goes through the points where the maximaare achieved.(b) On Figure 15.38, sketch the curve going through thepoints corresponding to the same maximum valuesin part (a). This curve is called the envelope of thecross-sections.(c) Show that the Lagrange multiplier for this constrainedoptimization problem is the slope of the envelopecurve in part (b).