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To find the gradient of the function ( f(x, y, z) = x^2 + y^2 + z^2 ), we first compute the partial derivatives with respect to ( x ), ( y ), and ( z ).
1. The partial derivative with respect to ( x ):
[
frac{partial f}{partial x} = 2x
]
2. The partial derivative with respect to ( y ):
[
frac{partial f}{partial y} = 2y
]
3. The partial derivative with respect to ( z ):
[
frac{partial f}{partial z} = 2z
]
Next, we evaluate these partial derivatives at the point ( (1, 1, 1) ):
– At ( (1, 1, 1) ):
[
frac{partial f}{partial x}(1, 1, 1) = 2 cdot 1 = 2
]
[
frac{partial f}{partial y}(1, 1, 1) = 2 cdot 1 = 2
]
[
frac{partial f}{partial z}(1, 1, 1) = 2 cdot 1 = 2
]
Answer: b
Explanation: Grad(x2+y2+z2) = 2xi + 2yj + 2zk. Put x=1, y=1, z=1, the gradient will be 2i +2j + 2k.