Question

Part b question D incorrect

Part a. Let
F(x, y, z)=9 z/1+x y⁴
If ∇ F(2,1,2)=(a, b, c) then
[ a=__; b=___; c=___ ]

Answer 1

To find the values of a, b, and c given ∇ F(2,1,2) = (a, b, c), calculate the partial derivatives of F(x, y, z) with respect to x, y, and z. Substituting the values (2,1,2) into the partial derivatives, we find a = 0, b = -144/81, and c = 9/5.

Step-by-step explanation:

To find the values of a, b, and c given ∇ F(2,1,2) = (a, b, c), we need to find the partial derivatives of F(x, y, z) with respect to x, y, and z. Let's calculate the partial derivatives:

1. Calculate ∂F/∂x: Differentiating F(x, y, z) = 9z/(1 + xy⁴) with respect to x, we get ∂F/∂x = 0.
2. Calculate ∂F/∂y: Differentiating F(x, y, z) = 9z/(1 + xy⁴) with respect to y, we get ∂F/∂y = -36xy³z/(1 + xy⁴)².
3. Calculate ∂F/∂z: Differentiating F(x, y, z) = 9z/(1 + xy⁴) with respect to z, we get ∂F/∂z = 9/(1 + xy⁴).

Substituting the values (2,1,2) into the partial derivatives, we have:

a = ∂F/∂x(2,1,2) = 0

b = ∂F/∂y(2,1,2) = -36(2)(1)(2)/(1 + (2)(1)⁴)² = -144/81

c = ∂F/∂z(2,1,2) = 9/(1 + (2)(1)⁴) = 9/5