Given f(x, y, z) = xy- yz + z. The gradient of f(x, y, z) is...

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Given f(x, y, z) = xy- yz + z. The gradient of f(x, y, z) is Vf(x, y, z) = (p,q, s), where 2xy x^2-z^3 -3yz^2+1 p S= (Input functions in increasing alphabetic order, input the constant term before any variable terms. E.g., y53 +2 should be input as 2-x^4+3y^5) Find the directional derivative of this f(x, y, z) at (1, -2, 0) in the direction of = (2, 1, -2): First, you evaluate V(1, -2, 0) = (| -4,1,1 J). Then, since is not unit, you need to normalize v. Eventually, you compute the directional derivative of this f(x, y, z) at (1, -2, 0) in the given direction to be Now study the rate of change only at the point (1,-2,0): -3 X 2/3 1/3 (1) If the maximum rate of change of f(x, y, z) happens in the direction (a, b, 6), then it can only be that a = X 3*(1/2) The maximum rate of change is always (for square root, use ^(1/2). E.g., 11 should be input as 11^(1/2). Depending on what method you use, some of you may need to simplify your result first, e.g., for 2/2, it is 2, so you only need to input 2^(1/2)) X -2/3 X-3 (2) If the minimum rate of change of f(x, y, z) happens in the direction (c, -5, d), then it can only be that c = .d= 3*(1/2) The minimum rate of change is always (for square root, use ^(1/2), e.g.. -11 should be input as -11^(1/2)) Given f(x, y, z) = xy- yz + z. The gradient of f(x, y, z) is Vf(x, y, z) = (p,q, s), where 2xy x^2-z^3 -3yz^2+1 p S= (Input functions in increasing alphabetic order, input the constant term before any variable terms. E.g., y53 +2 should be input as 2-x^4+3y^5) Find the directional derivative of this f(x, y, z) at (1, -2, 0) in the direction of = (2, 1, -2): First, you evaluate V(1, -2, 0) = (| -4,1,1 J). Then, since is not unit, you need to normalize v. Eventually, you compute the directional derivative of this f(x, y, z) at (1, -2, 0) in the given direction to be Now study the rate of change only at the point (1,-2,0): -3 X 2/3 1/3 (1) If the maximum rate of change of f(x, y, z) happens in the direction (a, b, 6), then it can only be that a = X 3*(1/2) The maximum rate of change is always (for square root, use ^(1/2). E.g., 11 should be input as 11^(1/2). Depending on what method you use, some of you may need to simplify your result first, e.g., for 2/2, it is 2, so you only need to input 2^(1/2)) X -2/3 X-3 (2) If the minimum rate of change of f(x, y, z) happens in the direction (c, -5, d), then it can only be that c = .d= 3*(1/2) The minimum rate of change is always (for square root, use ^(1/2), e.g.. -11 should be input as -11^(1/2))