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Consider a function f(x, y, z) = x2yz + 3xy2. The greatest rate of increase of function f at point (2, 1, -1) is
Correct answer is '9'. Can you explain this answer?
Verified Answer
Consider a function f(x, y, z) = x2yz + 3xy2. The greatest rate of inc...
Greatest rate of increase of function f is directional derivative at that point.
∇f = (2xyz + 3y2) i + (x2yz + 6xy) j + x2y k
At the given point (2, 1, -1), ∇f = -i + 8 j + 4 k
Greatest rate of increase
∇f = (2xyz + 3y2) i + (x2yz + 6xy) j + x2y k
At the given point (2, 1, -1), ∇f = -i + 8 j + 4 k
Greatest rate of increase
Most Upvoted Answer
Consider a function f(x, y, z) = x2yz + 3xy2. The greatest rate of inc...
Understanding the Function
The function in question is f(x, y, z) = x^2yz + 3xy^2. To find the greatest rate of increase, we need to calculate the gradient of the function.
Gradient Calculation
- The gradient ∇f is given by the vector of partial derivatives:
- ∂f/∂x = 2xyz + 3y^2
- ∂f/∂y = x^2z + 6xy
- ∂f/∂z = x^2y
Evaluating the Gradient at (2, 1, -1)
Now, we evaluate the gradient at the point (2, 1, -1):
- ∂f/∂x at (2, 1, -1):
- = 2(2)(1)(-1) + 3(1)^2 = -4 + 3 = -1
- ∂f/∂y at (2, 1, -1):
- = (2)^2(-1) + 6(2)(1) = -4 + 12 = 8
- ∂f/∂z at (2, 1, -1):
- = (2)^2(1) = 4
Thus, the gradient at (2, 1, -1) is ∇f(2, 1, -1) = (-1, 8, 4).
Magnitude of the Gradient
The greatest rate of increase of the function is the magnitude of the gradient:
- |∇f| = √((-1)^2 + (8)^2 + (4)^2)
- = √(1 + 64 + 16)
- = √81
- = 9
Conclusion
The greatest rate of increase of the function f at the point (2, 1, -1) is indeed 9.
The function in question is f(x, y, z) = x^2yz + 3xy^2. To find the greatest rate of increase, we need to calculate the gradient of the function.
Gradient Calculation
- The gradient ∇f is given by the vector of partial derivatives:
- ∂f/∂x = 2xyz + 3y^2
- ∂f/∂y = x^2z + 6xy
- ∂f/∂z = x^2y
Evaluating the Gradient at (2, 1, -1)
Now, we evaluate the gradient at the point (2, 1, -1):
- ∂f/∂x at (2, 1, -1):
- = 2(2)(1)(-1) + 3(1)^2 = -4 + 3 = -1
- ∂f/∂y at (2, 1, -1):
- = (2)^2(-1) + 6(2)(1) = -4 + 12 = 8
- ∂f/∂z at (2, 1, -1):
- = (2)^2(1) = 4
Thus, the gradient at (2, 1, -1) is ∇f(2, 1, -1) = (-1, 8, 4).
Magnitude of the Gradient
The greatest rate of increase of the function is the magnitude of the gradient:
- |∇f| = √((-1)^2 + (8)^2 + (4)^2)
- = √(1 + 64 + 16)
- = √81
- = 9
Conclusion
The greatest rate of increase of the function f at the point (2, 1, -1) is indeed 9.
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