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Unformatted text preview: OPMT 5701 Lecture Notes One Variable Optimization October 10, 2006 Critical Points A critical point occurs whenever the &rst derivative of a function is equal to zero, i.e. if y = f ( x ) then dy dx = f ( x ) = 0 is a critical point. A critical is a stationary value of the function. A critical point can be a. some kind of maximum point, b. some kind of minimum point, c. an inection point. Both a maximum and a minimum point can be a relative or global max or min (extremum). An inection point is neither a maximum or a minimum. graphically, an inection point is like a "shelf". All types of critical points are illustrated in &gure one. The First Derivative Test for relative Extremum The condition dy dx = f ( x ) = 0 is a necessary, but not su cient, condition to establish an extremum (max or min). In order to establish whether a critical point is an extremum we can use the following test: If, at x = x , f ( x ) = 0 , then f ( x ) will be (a) a relative Maximum if f ( x ) > for x < x f ( x ) < for x > x 1 x y = f(x) A B C D E Relative Max Inflection Relative Min Critical Points: f (x) = 0 (global Max) (global Min) (b) a relative Minimum if f ( x ) < for x < x f ( x ) > for x > x (c) an In&ection point if f ( x ) has the SAME SIGN for x < x and x > x Example: Suppose a revenue function is given by R = 10 x & x 2 The &rst order condition is R = 10 & 2 x = 0 x = 5 Applying the &rst derivative test, we see that, for x < 5 , 10 & 2 x > and, for x > 5 , 10 & 2 x < . Therefore we have a maximum. More on In&ection Points While f ( x ) = 0 may be an inection point, not all inection points occur where the &rst derivative is zero. Figure two illustrates two types of inection points. Also, each functions derivative is graphed directly below. Note the shape of each of the derivative functions: the &rst is...
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