Given the following function

𝑓
(
𝑥
,
𝑦
)
=
3
𝑥
2
+
𝑦
2
+
1

The gradient of f at (x,y) = (1,1) is

Question

Given the following function

𝑓
(
𝑥
,
𝑦
)
=
3
𝑥
2
+
𝑦
2
+
1

The gradient of f at (x,y) = (1,1) is

BlackTom AI · Accepted Answer

Question restatement: We’re given f(x,y) = 3x^2 + y^2 + 1, and asked for the gradient of f at the point (x, y) = (1, 1).

Option 1: [ 6 2 ]
- To find the gradient, compute partial derivatives: ∂f/∂x = 6x and ∂f/∂y = 2y. Evaluating at (1,1) gives ∂f/∂x = 6(1) = 6 and ∂f/∂y = 2(1) = 2. Therefore, ∇f(1,1) = (6, 2), which matches this option.

Option 2: [ 5 1 ]
- This would correspond to partial derivatives evaluated as ∂f/∂x = 5 and ∂f/∂y = 1 at (1,1). However, from the formulas ∂f/∂x = 6x and ∂f/∂y = 2y, substituting x = 1 and y = 1 yields 6 and 2, not 5 and 1. Hence this is incorrect.

Option 3: [ 4 1 ]
- This option proposes ∂f/∂x = 4 and ∂f/∂y = 1 at (1,1). Again, using the correct derivatives ∂f/∂x = 6x and ∂f/∂y = 2y with x = 1, y = 1 gives 6 and 2. The values 4 and 1 do not align with the actual partial derivatives at that point, so this is not correct.

Option 4: [ 7 2 ]
- This would imply ∂f/∂x = 7 and ∂f/∂y = 2 at (1,1). While the second component matches the actual ∂f/∂y = 2, the first component is off because ∂f/∂x at (1,1) should be 6, not 7. Therefore this choice is incorrect.

Summary of reasoning across options: The gradient components come directly from the partial derivatives ∂f/∂x = 6x and ∂f/∂y = 2y, which at (1,1) yield (6, 2). Any option differing in the first component from 6 or the second from 2 is inconsistent with the computed gradient. The only option that matches (6, 2) is the one showing 6 and 2 in the correct order.

BN5206 Re-opened Canvas questions [NON-GRADED, FOR PRACTICE]

Given the following function 𝑓 ( 𝑥 , 𝑦 ) = 3 𝑥 2 + 𝑦 2 + 1 The gradient of f at (x,y) = (1,1) is

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