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Answer :
The gradient vector ∇f(x, y, z) of the function f(x, y, z) = xy² - √5xz + 69 at the point (-4, 5, -3) is: ∇f(-4, 5, -3) = (25 - √15, -40, 4√5)
To compute the gradient vector ∇f(x, y, z) of the given function f(x, y, z) = xy² - √5xz + 69 at the point (-4, 5, -3), we need to find the partial derivatives of the function with respect to x, y, and z, and then evaluate them at the given point.
The gradient vector is defined as:
∇f(x, y, z) = (∂f/∂x, ∂f/∂y, ∂f/∂z)
Step 1: Find the partial derivatives of the function.
∂f/∂x = y² - √5z
∂f/∂y = 2xy
∂f/∂z = -√5x
Step 2: Evaluate the partial derivatives at the point (-4, 5, -3).
∂f/∂x = 5² - √5(-3) = 25 - √15
∂f/∂y = 2(-4)(5) = -40
∂f/∂z = -√5(-4) = 4√5
Step 3: Construct the gradient vector.
∇f(x, y, z) = (∂f/∂x, ∂f/∂y, ∂f/∂z)
∇f(-4, 5, -3) = (25 - √15, -40, 4√5)
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