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Section 15.1 Local Extrema

Section 15.1 Local Extrema. Definition of Local Extrema. Let P be a point in the domain of f . Then f has a local maximum at the point P 0 if f ( P 0 ) ≥ f ( P ) for all points P near P 0

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Section 15.1 Local Extrema

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  1. Section 15.1Local Extrema

  2. Definition of Local Extrema • Let P be a point in the domain of f. Then • f has a local maximum at the point P0 if f(P0 ) ≥ f(P) for all points P near P0 • f has a local minimum at the point P0 if f(P0 ) ≤ f(P) for all points P near P0 • These definitions are similar to those we saw back in single variable calculus • How did we determine local extrema back then?

  3. Definition of Critical Points • Suppose a function has a local maximum at P0 which does not lie on the boundary of the domain • Recall that the gradient vector (if it is defined and nonzero) points in the direction of maximum increase • What is direction of the maximum rate of increase at P0? • Thus we have critical points where the gradient is the zero vector or is undefined

  4. Definition of Critical Points • P0 is a critical point of f if

  5. Example • Find the critical points of the following functions • We will use maple to determine what is going on at these critical points • We need a test for determining what’s going on at our critical points

  6. Second Derivative Test • Old Stuff: Suppose f has a critical point at x = a, continuous first and second derivatives (at x = a) and f(a) = 0, then • Using a 2nd degree Taylor polynomial we have • Which is a quadratic whose concavity depends on the sign of f’’(a)

  7. Second Derivative Test for y = f(x) • Suppose f has a critical point at x = a, continuous first and second derivatives (at x = a) and f(a) = 0, then • f’’(a) > 0 → f(a) is a local minimum • f’’(a) < 0 → f(a) is a local maximum • f’’(a) = 0 → no conclusion can be drawn

  8. Second Derivative Test for Functions of Two Variables • Suppose Q is defined by • Then • Note

  9. Based on this form, what do you think we need in order to have our graph be concave up in the x direction? y direction?

  10. Given • If a > 0 and 4ac-b2 > 0 → Q has a local min at (0,0) • If a < 0 and 4ac-b2 > 0 → Q has a local max at (0,0) • If a > 0 and 4ac-b2 < 0 → Q has a saddle point at (0,0) • If a < 0 and 4ac-b2 < 0 → Q has a saddle point at (0,0) • If 4ac-b2 = 0 no conclusion can be drawn

  11. Second Derivative Test for Functions of Two Variables • Suppose f(x,y) has continuous 1st and 2nd parital derivatives at (0,0), that (0,0) is a critical point and f(0,0) = 0. From 14.7 we know

  12. Given • we have • So • Thus we have our discriminant and we can restate our previous conclusions

  13. Second Derivative Test for Functions of Two Variables • If f has a critical point at P0 then the discriminant is • If D > 0 and • fxx(P0) > 0 → f(P0) is a local minimum • fxx(P0) < 0 → f(P0) is a local maximum • If D < 0 →f(P0) is a saddle point • If D = 0 → no conclusion can be drawn from the second derivative test • Note: Since we solved for this test by completing the square in terms of x, we use the second partial of x

  14. Examples • Classify the critical points you found earlier

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