Final answer:
The maximum rate of change of the function f(x, y, z) = (6x+5y)/z at the point (6, 7, -1) is the magnitude of the gradient vector at that point, which is sqrt(5990). The direction vector of this rate of change is <-6/\sqrt{5990}, -5/\sqrt{5990}, 77/\sqrt{5990}>.
Step-by-step explanation:
The maximum rate of change of a function of several variables, like f(x, y, z) = \frac{6x+5y}{z} at a point, is given by the magnitude of the gradient of f at that point. To find this, we first need to calculate the partial derivatives of f with respect to x, y, and z.
The partial derivative with respect to x is \frac{6}{z}, with respect to y is \frac{5}{z}, and with respect to z is -\frac{6x+5y}{z^2}. At the point (6, 7, -1), these derivatives are -6, -5, and 77 respectively.
To find the direction of the maximum rate of change, we use the gradient vector, \\abla f, which is <-6, -5, 77> at the point (6, 7, -1). The magnitude of this vector is the maximum rate of change and is calculated as \sqrt{(-6)^2 + (-5)^2 + 77^2}, which simplifies to \sqrt{36 + 25 + 5929} = \sqrt{5990}.
The direction vector of the maximum rate of change is the normalized gradient vector <-6/\sqrt{5990}, -5/\sqrt{5990}, 77/\sqrt{5990}>, which points in the direction where the function increases most rapidly.