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# critical - CRITICAL POINTS Math21a O Knill HOMEWORK 13.8 2...

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Unformatted text preview: CRITICAL POINTS Math21a, O. Knill HOMEWORK: 13.8: 2, 18, 20, 30, 40 CRITICAL POINTS. A point ( x ,y ) in the plane is called a critical point of f ( x,y ) if ∇ f ( x ,y ) = (0 , 0). Critical points are also called stationary points . Critical points are candidates for extrema because at critical points, all directional derivatives D v f = ∇ f · v are zero. EXAMPLE 1. f ( x,y ) = x 4 + y 4 − 4 xy +2. The gradient is ∇ f ( x,y ) = (4( x 3 − y ) , 4( y 3 − x )) with critical points (0 , 0) , (1 , 1) , ( − 1 , − 1). EXAMPLE 2. f ( x,y ) = sin( x 2 + y ) + y . The gradient is ∇ f ( x,y ) = (2 x cos( x 2 + y ) , cos( x 2 + y ) + 1). For a critical points, we must have x = 0 and cos( y ) + 1 = 0 which means π + k 2 π . The critical points are at (0 ,π ) , (0 , 3 π ) ,... . EXAMPLE 3. (”volcano”) f ( x,y ) = ( x 2 + y 2 ) e − x 2 − y 2 . The gradient ∇ F = (2 x − 2 x ( x 2 + y 2 ) , 2 y − 2 y ( x 2 + y 2 )) e − x 2 − y 2 vanishes at (0 , 0) and on the circle x 2 + y 2 = 1. There are ∞ many critical points. EXAMPLE 4 (”pendulum”) f ( x,y ) = − g cos( x ) + y 2 / 2 is the energy of the pendulum. The gradient ∇ F = ( y, − g sin( x )) is (0 , 0) for x = 0 ,π, 2 π,...,y = 0. These points are equilibrium points, where the pendulum is at rest. EXAMPLE 5 (”Volterra Lodka”) f ( x,y ) = a log( y ) − by + c log( x ) − dx . (This function is left invariant by the flow of the Volterra Lodka differential equation ˙ x = ax − bxy, ˙ y = − cy + dxy which you might have seen in Math1b.) The point ( c/d,a/b ) is a critical point.) is a critical point....
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