Question at position 1 If f(x,y,z)=x2yz2+xy2z+xyf\left(x,y,z\right)=x^2yz^2+xy^2z+xy, then fx(1, 2, 3) =36.none of the above55.50.48.

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Question at position 1 If f(x,y,z)=x2yz2+xy2z+xyf\left(x,y,z\right)=x^2yz^2+xy^2z+xy, then fx(1, 2, 3) =36.none of the above55.50.48.

BlackTom AI · Accepted Answer

We start by parsing the function and the request. The function given is f(x,y,z) = x^2 y z^2 + x y^2 z + x y, and we are asked to compute the partial derivative with respect to x, denoted fx, at the point (1, 2, 3). The approach is to differentiate term by term with respect to x, treating y and z as constants.

Option A: 36.
- If someone only differentiates the first term x^2 y z^2 with respect to x, they would get 2x y z^2. Plugging in x=1, y=2, z=3 gives 2(1)(2)(3^2) = 2*2*9 = 36. This misses the contributions from the other x-dependent terms, so this value is incomplete.

Option B: none of the above
- This would be true only if the computed fx at (1,2,3) did not match any listed option. Since fx has multiple terms and we must check all options, we cannot conclude this without finishing the full calculation. At this stage, we should compare the full evaluation to all options later.

Option C: 55.
- This would require the sum of the x-derivatives of all terms to equal 55. However, the full derivative includes three components (as shown below). If we mis-sum or omit a term, we might land on 55 by error, but with correct calculation the total will be the sum of all contributions, not a single term, so 55 would only be correct if the total happened to be 55, which we will verify.

Option D: 50.
- This aligns with the complete evaluation when all x-dependent derivatives are included. The full fx comes from differentiating each term:
  - d/dx(x^2 y z^2) = 2x y z^2
  - d/dx(x y^2 z) = y^2 z
  - d/dx(x y) = y
  Therefore fx = 2x y z^2 + y^2 z + y. Substituting x=1, y=2, z=3 gives 2(1)(2)(3^2) + (2^2)(3) + 2 = 2*2*9 + 4*3 + 2 = 36 + 12 + 2 = 50. This is the complete and correct evaluation for fx at the given point.

Option E: 48.
- This would correspond to missing one of the terms in the full derivative, such as omitting the y term or the y^2 z term, or possibly miscalculating z^2 as 8 or some other arithmetic slip. However, with the correct formula fx = 2x y z^2 + y^2 z + y and the given point, the total is 50, not 48.

Summary of reasoning across options:
- The complete derivative fx includes three components: 2x y z^2, y^2 z, and y. Evaluating at (1,2,3) yields 36 from the first component, 12 from the second, and 2 from the third, summing to 50. Any option that only accounts for one component (like 36) is incomplete. Any option that asserts a value other than 50 must arise from an arithmetic error or an omission. The option claiming none of the above would be incorrect given the full correct value, while 55 and 48 do not match the full computed sum. The careful, term-by-term derivative leads to the total of 50, not 36, 55, or 48, and not “none of the above.”

Question at position 1 If f(x,y,z)=x2yz2+xy2z+xyf\left(x,y,z\right)=x^2yz^2+xy^2z+xy, then fx(1, 2, 3) =36.none of the above55.50.48.

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