$4.2 Extrema & $4.3 The Mean Value Theorem & Monotonicity (8) Find all critical points of the functions below. Determine whether the critical point is a local max, local min, or neither. (a) f(x) = x], on R (b) f(x) = 23/5, on IR (c) f(x) = 23/5, on [1, 2] (9) Find the local and global extrema of a4 - 2x2 on [-2, 3]. (10) find a point c satisfying the conclusion of the MVT for the given function and interval where f (x) = =, on [1, 2] $4.4 The Second Derivative and Concavity (11) Determine the intervals on which the function y = x2 - 4x + 9 is concave up or down and find the points of inflection. (12) Find the critical points and apply the Second Derivative Test (or state that it fails). (a) f(x) = 23 - 12x2 + 45x, (b) f(x) = 673/2 - 4x1/2, (13) Following (2), find the transition points, intervals of increase/decrease, concavity, and asymptotic behavior of the functions below. Then sketch the graph, with this information indicated. (a) f(x) = 23 - 12x2 + 45x, (b) f(x) = 62-3/2 - 4x1/2,HOMEWORK 6 $3.5 - $4.4 $3.5 Higher Derivatives (1) Find y" and y"" (a) y(a) = x -x2+1, (b) y(x) = vo+ (c) y(x) = 1-x $3.7 The Chain Rule (2) Write the function as a composite f(g(x)) and compute the derivative using the Chain Rule. (a) f(x) = Vx2+1, (b) f (x) = (x2+1)-2, $3.8 Implicit Differentiation (3) Verify that the given point lies on the contour given by the relatin, then find at the given point below: (2 + 2)2 -6(2y + 3)2 = 3, at (1, -1), (4) Find all points on the graph of 3x2 + 4y' + 3ry = 24 where the tangent line is horizontal. $4.1 Linea Approximation (5) The cube root of 64 is 4. How much smaller is the cube root of 63.8? Estimate using the Linear Approxi- mation. (6) Use the Linear Approximation to estimate f(a + Ar), where f(x) = vx + 1, a = 8, Ax = 0.1 (7) Find the linearization at a and then use it to approximate f(b). f(x) = v23 + 1, a = 2 and b = 2.1