MAT 21C Lecture Notes - Lecture 25: Maxima And Minima, Quadratic Function, Saddle Point
MAT 21C – Lecture 25 –Extreme Values of Multi-Variable Functions
• If is a continuous function on a closed, bounded domain R in the plane,
then attains its maximum and minimum values at some points in R.
• Possible points where attains maximum and/or minimums:
o a) Critical points: interior points of R such that either i) or is not
defined or ii)
o b) Boundary points of R
• We narrow our focus to only critical points and boundary points in the search for
maximums and minimums.
• Second Derivative Test: If for f(x) in which x = a is a critical point, then
o a) If , then there is a local minimum at x = a
o b) If , then there is a local maximum at x = a
o c) If , then the test is inconclusive.
• The discriminant is defined as
.
• Suppose (a, b) is a critical point of such that .
o a) If and , then there is a local maximum at (a, b)
o b) If and , then there is a local minimum at (a, b)
o c) If , then there is a saddle point at (a, b)
o d) If , then no conclusion can be made.
• Examples:
o 1) Determine if any local extrema exist for .
The critical point(s) of f are the solutions to
The only critical point of f
is (0, 0). Applying the discriminant, we obtain
. Since and , then there is a local
minimum of .
o 2) Given , the critical point(s) of f are solutions to
Critical point of f is (0, 0). Then
Since and , then
there is a local maximum of .
o 3) Find the local extrema, if any, for
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Document Summary
Mat 21c lecture 25 extreme values of multi-variable functions: b) boundary points of r maximums and minimums, we narrow our focus to only critical points and boundary points in the search for. The critical point(s) of f are the solutions to (cid:4666)(cid:1876)(cid:2870)+(cid:1877)(cid:2870)(cid:4667)= (cid:3643)< (cid:884)(cid:1877)=(cid:882)}(cid:3643){(cid:1876)=(cid:882) (cid:884)(cid:1876),(cid:884)(cid:1877)> = (cid:3643){(cid:884)(cid:1876)=(cid:882) (cid:1877)=(cid:882)}. The only critical point of f is (0, 0). Applying the discriminant, we obtain =|(cid:3051)(cid:3051) (cid:3051)(cid:3052) (cid:3052)(cid:3051) (cid:3052)(cid:3052)|=|(cid:884) (cid:882)(cid:882) (cid:884)|= (cid:884)(cid:4666)(cid:884)(cid:4667) (cid:882)(cid:2870)=(cid:886)>(cid:882). Since >(cid:882) and (cid:3051)(cid:3051)>(cid:882), then there is a local minimum of (cid:4666)(cid:882),(cid:882)(cid:4667)=(cid:882): 2) given (cid:4666)(cid:1876),(cid:1877)(cid:4667)= (cid:4666)(cid:1876)(cid:2870)+(cid:1877)(cid:2870)(cid:4667), the critical point(s) of f are solutions to. Since >(cid:882) and (cid:3051)(cid:3051) (cid:3643) =
