The function given is f(x, y, z) = x + z^2. To find the directions in which the function increases and decreases most rapidly at the point P(1, 1, 2, 1/2), we need to find the gradient of the function.

The gradient of a function is a vector that points in the direction of the greatest rate of increase of the function, and its magnitude is the rate of increase in that direction.

The gradient of the function f is given by the vector of its partial derivatives with respect to each variable. So, we first need to find these partial derivatives.

The partial derivative of f with respect to x is 1, with respect to y is 0 (since y does not appear in the function), and with respect to z is 2z.

So, the gradient of f is the vector (1, 0, 2z).

At the point P(1, 1, 2, 1/2), the z-coordinate is 1/2, so the gradient of f at P is (1, 0, 2*1/2) = (1, 0, 1).

This means that the function f increases most rapidly in the direction of the vector (1, 0, 1) and decreases most rapidly in the opposite direction, which is given by the vector (-1, 0, -1).

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The function given is f(x, y, z) = x + z^2. To find the directions in which the function increases and decreases most rapidly at the point P(1, 1, 2, 1/2), we need to find the gradient of the function.

The gradient of a function is a vector that points in the direction of the greatest rate of increase of the function, and its magnitude is the rate of increase in that direction.

The gradient of the function f is given by the vector of its partial derivatives with respect to each variable. So, we first need to find these partial derivatives.

The partial derivative of f with respect to x is 1, with respect to y is 0 (since y does not appear in the function), and with respect to z is 2z.

So, the gradient of f is the vector (1, 0, 2z).

At the point P(1, 1, 2, 1/2), the z-coordinate is 1/2, so the gradient of f at P is (1, 0, 2*1/2) = (1, 0, 1).

This means that the function f increases most rapidly in the direction of the vector (1, 0, 1) and decreases most rapidly in the opposite direction, which is given by the vector (-1, 0, -1).

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The function given is f(x, y, z) = x + z^2. To find the directions in which the function increases and decreases most rapidly at the point P(1, 1, 2, 1/2), we need to find the gradient of the function.

The gradient of a function is a vector that points in the direction of the greatest rate of increase of the function, and its magnitude is the rate of increase in that direction.

The gradient of the function f is given by the vector of its partial derivatives with respect to each variable. So, we first need to find these partial derivatives.

The partial derivative of f with respect to x is 1, with respect to y is 0 (since y does not appear in the function), and with respect to z is 2z.

So, the gradient of f is the vector (1, 0, 2z).

At the point P(1, 1, 2, 1/2), the z-coordinate is 1/2, so the gradient of f at P is (1, 0, 2*1/2) = (1, 0, 1).

This means that the function f increases most rapidly in the direction of the vector (1, 0, 1) and decreases most rapidly in the opposite direction, which is given by the vector (-1, 0, -1).

Find the directions in which the function f (x, y, z) = sett + z2 increases and decreasesmost rapidly at the point P(1,1n 2,1/2)

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