2. Find and classify all critical points for f(x,y) = -22 + y? (x - 8)
(8 points) Find all critical points and classify them via the second derivative test. (a) f(x,y) = 2.ry+y – 3y - 2 (b) f(x,y) = ye" – y? - I
9y3 + 3x2y-6y + 2 . 3. Find and classify all the critical points of f(x,y) 9y3 + 3x2y-6y + 2 . 3. Find and classify all the critical points of f(x,y)
[8 points) For the curve given by x = t4 – t, y = Int + t2, find the equation of the tangent line to the curve at the point (0,1).
Consider the nonlinear system ?x′ = ln(y^2 − x) and y'=x-y-1 (a)Find all the critical points (b)Find the corresponding linearized system near the critical points. (c) Classify the (i) type (node, saddle point, · · · ), and (ii) stability of the critical points for the corresponding linearized system. (d) What conclusion can you obtain for the type and stability of the critical points for the original nonlinear system?
8. Let y = x2 cos x, Find y' 9. Let g(x) = -2 cos x, Find g'(x) 10. Find F(x) = (4x + 3)5, Find F'(x) BONUS QUESTION (15 POINTS Let y = (4x - 3)(x - 1)5; Find y"
(a) Find and classify all of the critical points of the function X f(x, y, z) = (x2 +42 + x2)3/2 on the unit sphere. (b) Find and classify all of the critical points of the function f(x, y, z) = x sin(x2 + y2 +22) on the sphere of radius
PROBLEM 7 Find all critical points of g(x, y) = 12 y - 6xy - 63/2x + 144x + y2 + 3x - 6. Input your answer as a sequence of one or more ordered pairs separated by commas. For example: (1,2), (3,4), (5,6)
5. Let f(x,y) = 3x2 y - y3 - 6x. (a) Find all the critical points off. (b) Classify each of the critical points; that is, what type are they? (c) For the same function f(x,y), find the maximum value of f on the unit square, 0 SX S1,0 <y s 1.
QUESTION 7 Find all the critical points for f(x,y)=-x® + 3x - xy and classify each as a local maximum, local minimum or a saddle point. (9 marks)