The directional derivative of a function f(x,y,z) is maximized in the direction of the gradient vector f. First, we need to calculate the partial derivatives of f(x,y,z) = x²yz³ with respect to x, y, and z.

The directional derivative of a function f(x,y,z) is maximized in the direction of the gradient vector f. First, we need to calculate the partial derivatives of f(x,y,z) = x²yz³ with respect to x, y, and z.

We compute the partial derivatives: ( f)/( x) by treating y and z as constants, ( f)/( y) by treating x and z as constants, and ( f)/( z) by treating x and y as constants.

Find the maximum value of the directional derivative of f(x,y,z) = x²yz³ at point (1,1,1) in any direction.

Answer: The maximum value of the directional derivative is $\sqrt{14}$ .

Step-by-step solution

Step 1: Calculate the gradient of the function

The directional derivative of a function $f(x,y,z)$ is maximized in the direction of the gradient vector $\nabla f$ . First, we need to calculate the partial derivatives of $f(x,y,z) = x^2yz^3$ with respect to $x$ , $y$ , and $z$ .

Step 2: Compute partial derivatives

We compute the partial derivatives: $\frac{\partial f}{\partial x}$ by treating $y$ and $z$ as constants, $\frac{\partial f}{\partial y}$ by treating $x$ and $z$ as constants, and $\frac{\partial f}{\partial z}$ by treating $x$ and $y$ as constants.

Step 3: Evaluate the gradient at the given point

Now we substitute the point $(1{,}1{,}1)$ into the partial derivatives to find the gradient vector at that specific point. This vector indicates the direction of the steepest ascent of the function.

Step 4: Calculate the magnitude of the gradient

The maximum value of the directional derivative is the magnitude of the gradient vector. We calculate the magnitude using the formula for the length of a vector in three dimensions.

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