Question

What is the direction of fastest increase at \( (5,-4,6) \) for the function \( f(x, y, z)=\frac{1}{x^{2}+y^{2}+z^{2}} \). (Use symbolic notation and fractions where needed. Give your answer in the fo

Answer 1

The direction of fastest increase at the point (5,-4,6) for the function f(x, y, z) is given by the gradient of f evaluated at that point, which is \\abla f(5,-4,6) = \left(-\frac{10}{77^{2}}, \frac{8}{77^{2}}, -\frac{12}{77^{2}}\right).

Step-by-step explanation:

The direction of fastest increase for the function f(x, y, z)=\frac{1}{x^{2}+y^{2}+z^{2}} at the point (5,-4,6) can be found by calculating the gradient of f, denoted as \\abla f. The gradient of a function points in the direction of the greatest rate of increase of the function.

First, we compute the partial derivatives of f with respect to x, y, and z:

• 
  `(\partial f)/(\partial x) = -(2x)/((x^(2)+y^(2)+z^(2))^(2))\\(\partial f)/(\partial y) = -(2y)/((x^(2)+y^(2)+z^(2))^(2))\\(\partial f)/(\partial z) = -(2z)/((x^(2)+y^(2)+z^(2))^(2))`

Next, evaluate these derivatives at the point (5,-4,6):

• 
  `(\partial f)/(\partial x)|_((5,-4,6)) = -(2 \cdot 5)/((5^(2)+(-4)^(2)+6^(2))^(2))\\ ( \partial f)/(\partial y)|_((5,-4,6)) = -(2 \cdot (-4))/((5^(2)+(-4)^(2)+6^(2))^(2)) \\ (\partial f)/(\partial z)|_((5,-4,6)) = -(2 \cdot 6)/((5^(2)+(-4)^(2)+6^(2))^(2))`

Thus, the gradient of f at (5,-4,6) is:

\\abla f(5,-4,6) = \left(-\frac{10}{77^{2}}, \frac{8}{77^{2}}, -\frac{12}{77^{2}}\right)

Since the gradient points in the direction of fastest increase, \\abla f(5,-4,6) is the direction of fastest increase at the point (5,-4,6).

Answer 2

The direction of fastest increase at (5,-4,6) for the function f(x, y, z)= 1/x²+y²+z² is approximately -0.95i + 0.76j - 1.14k.

Step-by-step explanation:

To find the direction of the fastest increase at the point (5, -4, 6) for the function f(x, y, z) = 1/(x² + y² + z²), we can use the gradient vector. The gradient vector of a function points in the direction of the greatest rate of increase of the function at a given point.

The gradient vector ∇f(x, y, z) of a scalar function f(x, y, z) is given by:

∇f(x, y, z) = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k

Where i, j, and k are the unit vectors in the x, y, and z directions respectively.

For the function f(x, y, z) = 1/(x² + y² + z²), we have:

∂f/∂x = -2x/(x² + y² + z²)²

∂f/∂y = -2y/(x² + y² + z²)²

∂f/∂z = -2z/(x² + y² + z²)²

Evaluating these partial derivatives at the point (5, -4, 6), we get:

∂f/∂x = -2(5)/(5² + (-4)² + 6²)² = -10/841

∂f/∂y = -2(-4)/(5² + (-4)² + 6²)² = 8/841

∂f/∂z = -2(6)/(5² + (-4)² + 6²)² = -12/841

Therefore, the gradient vector ∇f(5,-4,6) is:

∇f(5,-4,6) = (-10/841)i + (8/841)j - (12/841)k

The direction of fastest increase is given by this gradient vector. To express this as a unit vector in the same direction, we divide each component by the magnitude of the gradient vector:

Magnitude of ∇f(5,-4,6):

|∇f(5,-4,6)| = sqrt((-10/841)² + (8/841)² + (-12/841)²)

≈ sqrt(0.0029)

Unit vector in the direction of fastest increase:

u = ((-10/841)/sqrt(0.0029))i + ((8/841)/sqrt(0.0029))j - ((12/841)/sqrt(0.0029))k ≈ -0.95i + 0.76j - 1.14k

Therefore, at the point (5,-4,6), the direction of fastest increase for the function f(x,y,z)=1/(x²+y²+z²) is approximately -0.95i + 0.76j - 1.14k.